α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
⊢ ∃ u x, Set.Countable u ∧ PairwiseDisjoint u B ∧ ↑↑μ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) = 0
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
this : ∀ (x : α), ∃ R, 0 < R ∧ R ≤ 1 ∧ ↑↑μ (closedBall x (20 * R)) < ⊤
⊢ ∃ u x, Set.Countable u ∧ PairwiseDisjoint u B ∧ ↑↑μ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) = 0
choose R hR0 hR1 hRμ using this
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
⊢ ∃ u x, Set.Countable u ∧ PairwiseDisjoint u B ∧ ↑↑μ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) = 0
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
⊢ ∃ u x, Set.Countable u ∧ PairwiseDisjoint u B ∧ ↑↑μ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) = 0
let t' := { a ∈ t | r a ≤ R (c a) }
case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
⊢ ∃ u x, Set.Countable u ∧ PairwiseDisjoint u B ∧ ↑↑μ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) = 0
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
⊢ ∃ u x, Set.Countable u ∧ PairwiseDisjoint u B ∧ ↑↑μ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) = 0
obtain ⟨u, ut', u_disj, hu⟩ : ∃ (u : _) (_ : u ⊆ t'),
    u.PairwiseDisjoint B ∧ ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b := by
  have A : ∀ a ∈ t', r a ≤ 1 := by
    intro a ha
    apply ha.2.trans (hR1 (c a))
  have A' : ∀ a ∈ t', (B a).Nonempty :=
    fun a hat' => Set.Nonempty.mono interior_subset (ht a hat'.1)
  refine' exists_disjoint_subfamily_covering_enlargment B t' r 2 one_lt_two (fun a ha => _) 1 A A'
  exact nonempty_closedBall.1 ((A' a ha).mono (hB a ha.1))
case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
⊢ ∃ u x, Set.Countable u ∧ PairwiseDisjoint u B ∧ ↑↑μ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) = 0
case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
⊢ ∃ u x, Set.Countable u ∧ PairwiseDisjoint u B ∧ ↑↑μ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) = 0
have ut : u ⊆ t := fun a hau => (ut' hau).1
case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
⊢ ∃ u x, Set.Countable u ∧ PairwiseDisjoint u B ∧ ↑↑μ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) = 0
case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
⊢ ∃ u x, Set.Countable u ∧ PairwiseDisjoint u B ∧ ↑↑μ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) = 0
have u_count : u.Countable := u_disj.countable_of_nonempty_interior fun a ha => ht a (ut ha)
case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
⊢ ↑↑μ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) = 0
case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
⊢ ∃ u x, Set.Countable u ∧ PairwiseDisjoint u B ∧ ↑↑μ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) = 0
refine' ⟨u, fun a hat' => (ut' hat').1, u_count, u_disj, _⟩
case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
⊢ ∃ u_1, u_1 ∈ 𝓝[s \ ⋃ (a : ι) (_ : a ∈ u), B a] x ∧ ↑↑μ u_1 = 0
case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
⊢ ↑↑μ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) = 0
refine' null_of_locally_null _ fun x _ => _
case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
⊢ ∃ u_1, u_1 ∈ 𝓝[s \ ⋃ (a : ι) (_ : a ∈ u), B a] x ∧ ↑↑μ u_1 = 0
case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
⊢ ∃ u_1, u_1 ∈ 𝓝[s \ ⋃ (a : ι) (_ : a ∈ u), B a] x ∧ ↑↑μ u_1 = 0
let v := { a ∈ u | (B a ∩ ball x (R x)).Nonempty }
case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
⊢ ∃ u_1, u_1 ∈ 𝓝[s \ ⋃ (a : ι) (_ : a ∈ u), B a] x ∧ ↑↑μ u_1 = 0
case intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
⊢ ∃ u_1, u_1 ∈ 𝓝[s \ ⋃ (a : ι) (_ : a ∈ u), B a] x ∧ ↑↑μ u_1 = 0
have vu : v ⊆ u := fun a ha => ha.1
case intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
⊢ ↑↑μ ((s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)) ≤ ε
case intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
⊢ ∃ u_1, u_1 ∈ 𝓝[s \ ⋃ (a : ι) (_ : a ∈ u), B a] x ∧ ↑↑μ u_1 = 0
refine' ⟨_ ∩ ball x (R x), inter_mem_nhdsWithin _ (ball_mem_nhds _ (hR0 _)),
  nonpos_iff_eq_zero.mp (le_of_forall_le_of_dense fun ε εpos => _)⟩
case intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
⊢ ↑↑μ ((s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)) ≤ ε
case intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
⊢ ↑↑μ ((s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)) ≤ ε
have I : (∑' a : v, μ (B a)) < ∞ := by
  calc
    (∑' a : v, μ (B a)) = μ (⋃ a ∈ v, B a) := by
      rw [measure_biUnion (u_count.mono vu) _ fun a ha => (h't _ (vu.trans ut ha)).measurableSet]
      exact u_disj.subset vu
    _ ≤ μ (closedBall x K) := (measure_mono (iUnion₂_subset fun a ha => hK a (vu ha) ha.2))
    _ < ∞ := μK
case intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
⊢ ↑↑μ ((s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)) ≤ ε
case intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
⊢ ↑↑μ ((s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)) ≤ ε
obtain ⟨w, hw⟩ : ∃ w : Finset v, (∑' a : { a // a ∉ w }, μ (B a)) < ε / C :=
  haveI : 0 < ε / C := by
    simp only [ENNReal.div_pos_iff, εpos.ne', ENNReal.coe_ne_top, Ne.def, not_false_iff,
      and_self_iff]
  ((tendsto_order.1 (ENNReal.tendsto_tsum_compl_atTop_zero I.ne)).2 _ this).exists
case intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
M : (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x) ⊆ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
⊢ ↑↑μ ((s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)) ≤ ε
case intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
⊢ ↑↑μ ((s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)) ≤ ε
have M : (s \ ⋃ a ∈ u, B a) ∩
    ball x (R x) ⊆ ⋃ a : { a // a ∉ w }, closedBall (c a) (3 * r a) := by
  intro z hz
  set k := ⋃ (a : v) (ha : a ∈ w), B a
  have k_closed : IsClosed k := isClosed_biUnion w.finite_toSet fun i _ => h't _ (ut (vu i.2))
  have z_notmem_k : z ∉ k := by
    simp only [not_exists, exists_prop, mem_iUnion, mem_sep_iff, forall_exists_index,
      SetCoe.exists, not_and, exists_and_right, Subtype.coe_mk]
    intro b hbv _ h'z
    have : z ∈ (s \ ⋃ a ∈ u, B a) ∩ ⋃ a ∈ u, B a :=
      mem_inter (mem_of_mem_inter_left hz) (mem_biUnion (vu hbv) h'z)
    simpa only [diff_inter_self]
  have : ball x (R x) \ k ∈ 𝓝 z := by
    apply IsOpen.mem_nhds (isOpen_ball.sdiff k_closed) _
    exact (mem_diff _).2 ⟨mem_of_mem_inter_right hz, z_notmem_k⟩
  obtain ⟨d, dpos, hd⟩ : ∃ d, 0 < d ∧ closedBall z d ⊆ ball x (R x) \ k :=
    nhds_basis_closedBall.mem_iff.1 this
  obtain ⟨a, hat, ad, rfl⟩ : ∃ a ∈ t, r a ≤ min d (R z) ∧ c a = z
  exact hf z ((mem_diff _).1 (mem_of_mem_inter_left hz)).1 (min d (R z)) (lt_min dpos (hR0 z))
  have ax : B a ⊆ ball x (R x) := by
    refine' (hB a hat).trans _
    refine' Subset.trans _ (hd.trans (diff_subset (ball x (R x)) k))
    exact closedBall_subset_closedBall (ad.trans (min_le_left _ _))
  obtain ⟨b, bu, ab, bdiam⟩ : ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
  exact hu a ⟨hat, ad.trans (min_le_right _ _)⟩
  have bv : b ∈ v := by
    refine' ⟨bu, ab.mono _⟩
    rw [inter_comm]
    exact inter_subset_inter_right _ ax
  let b' : v := ⟨b, bv⟩
  have b'_notmem_w : b' ∉ w := by
    intro b'w
    have b'k : B b' ⊆ k := @Finset.subset_set_biUnion_of_mem _ _ _ (fun y : v => B y) _ b'w
    have : (ball x (R x) \ k ∩ k).Nonempty := by
      apply ab.mono (inter_subset_inter _ b'k)
      refine' ((hB _ hat).trans _).trans hd
      exact closedBall_subset_closedBall (ad.trans (min_le_left _ _))
    simpa only [diff_inter_self, Set.not_nonempty_empty]
  let b'' : { a // a ∉ w } := ⟨b', b'_notmem_w⟩
  have zb : c a ∈ closedBall (c b) (3 * r b) := by
    rcases ab with ⟨e, ⟨ea, eb⟩⟩
    have A : dist (c a) e ≤ r a := mem_closedBall'.1 (hB a hat ea)
    have B : dist e (c b) ≤ r b := mem_closedBall.1 (hB b (ut bu) eb)
    simp only [mem_closedBall]
    linarith only [dist_triangle (c a) e (c b), A, B, bdiam]
  suffices H : closedBall (c b'') (3 * r b'') ⊆ ⋃ a : { a // a ∉ w }, closedBall (c a) (3 * r a)
  exact H zb
  exact subset_iUnion (fun a : { a // a ∉ w } => closedBall (c a) (3 * r a)) b''
case intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
M : (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x) ⊆ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
this : Encodable ↑v
⊢ ↑↑μ ((s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)) ≤ ε
case intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
M : (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x) ⊆ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
⊢ ↑↑μ ((s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)) ≤ ε
haveI : Encodable v := (u_count.mono vu).toEncodable
no goals
case intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
M : (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x) ⊆ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
this : Encodable ↑v
⊢ ↑↑μ ((s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)) ≤ ε
calc
  μ ((s \ ⋃ a ∈ u, B a) ∩ ball x (R x)) ≤ μ (⋃ a : { a // a ∉ w }, closedBall (c a) (3 * r a)) :=
    measure_mono M
  _ ≤ ∑' a : { a // a ∉ w }, μ (closedBall (c a) (3 * r a)) := (measure_iUnion_le _)
  _ ≤ ∑' a : { a // a ∉ w }, C * μ (B a) := (ENNReal.tsum_le_tsum fun a => μB a (ut (vu a.1.2)))
  _ = C * ∑' a : { a // a ∉ w }, μ (B a) := ENNReal.tsum_mul_left
  _ ≤ C * (ε / C) := (mul_le_mul_left' hw.le _)
  _ ≤ ε := ENNReal.mul_div_le
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
x : α
⊢ ∃ R, 0 < R ∧ R ≤ 1 ∧ ↑↑μ (closedBall x (20 * R)) < ⊤
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
⊢ ∀ (x : α), ∃ R, 0 < R ∧ R ≤ 1 ∧ ↑↑μ (closedBall x (20 * R)) < ⊤
intro x
case intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
x : α
R : ℝ
Rpos : 0 < R
μR : ↑↑μ (closedBall x R) < ⊤
⊢ ∃ R, 0 < R ∧ R ≤ 1 ∧ ↑↑μ (closedBall x (20 * R)) < ⊤
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
x : α
⊢ ∃ R, 0 < R ∧ R ≤ 1 ∧ ↑↑μ (closedBall x (20 * R)) < ⊤
obtain ⟨R, Rpos, μR⟩ : ∃ R, 0 < R ∧ μ (closedBall x R) < ∞ :=
  (μ.finiteAt_nhds x).exists_mem_basis nhds_basis_closedBall
case intro.intro.refine'_1
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
x : α
R : ℝ
Rpos : 0 < R
μR : ↑↑μ (closedBall x R) < ⊤
⊢ 0 < min 1 (R / 20)

case intro.intro.refine'_2
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
x : α
R : ℝ
Rpos : 0 < R
μR : ↑↑μ (closedBall x R) < ⊤
⊢ ↑↑μ (closedBall x (20 * min 1 (R / 20))) < ⊤
case intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
x : α
R : ℝ
Rpos : 0 < R
μR : ↑↑μ (closedBall x R) < ⊤
⊢ ∃ R, 0 < R ∧ R ≤ 1 ∧ ↑↑μ (closedBall x (20 * R)) < ⊤
refine' ⟨min 1 (R / 20), _, min_le_left _ _, _⟩
case intro.intro.refine'_1
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
x : α
R : ℝ
Rpos : 0 < R
μR : ↑↑μ (closedBall x R) < ⊤
⊢ 0 < R / 20
case intro.intro.refine'_1
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
x : α
R : ℝ
Rpos : 0 < R
μR : ↑↑μ (closedBall x R) < ⊤
⊢ 0 < min 1 (R / 20)
simp only [true_and_iff, lt_min_iff, zero_lt_one]
no goals
case intro.intro.refine'_1
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
x : α
R : ℝ
Rpos : 0 < R
μR : ↑↑μ (closedBall x R) < ⊤
⊢ 0 < R / 20
linarith
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
x : α
R : ℝ
Rpos : 0 < R
μR : ↑↑μ (closedBall x R) < ⊤
⊢ closedBall x (20 * min 1 (R / 20)) ⊆ closedBall x R
case intro.intro.refine'_2
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
x : α
R : ℝ
Rpos : 0 < R
μR : ↑↑μ (closedBall x R) < ⊤
⊢ ↑↑μ (closedBall x (20 * min 1 (R / 20))) < ⊤
apply lt_of_le_of_lt (measure_mono _) μR
case h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
x : α
R : ℝ
Rpos : 0 < R
μR : ↑↑μ (closedBall x R) < ⊤
⊢ 20 * min 1 (R / 20) ≤ R
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
x : α
R : ℝ
Rpos : 0 < R
μR : ↑↑μ (closedBall x R) < ⊤
⊢ closedBall x (20 * min 1 (R / 20)) ⊆ closedBall x R
apply closedBall_subset_closedBall
no goals
case h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
x : α
R : ℝ
Rpos : 0 < R
μR : ↑↑μ (closedBall x R) < ⊤
⊢ 20 * min 1 (R / 20) ≤ R
calc
  20 * min 1 (R / 20) ≤ 20 * (R / 20) :=
    mul_le_mul_of_nonneg_left (min_le_right _ _) (by norm_num)
  _ = R := by ring
no goals
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
x : α
R : ℝ
Rpos : 0 < R
μR : ↑↑μ (closedBall x R) < ⊤
⊢ 0 ≤ 20
norm_num
no goals
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
x : α
R : ℝ
Rpos : 0 < R
μR : ↑↑μ (closedBall x R) < ⊤
⊢ 20 * (R / 20) = R
ring
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
A : ∀ (a : ι), a ∈ t' → r a ≤ 1
⊢ ∃ u x, PairwiseDisjoint u B ∧ ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
⊢ ∃ u x, PairwiseDisjoint u B ∧ ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
have A : ∀ a ∈ t', r a ≤ 1 := by
  intro a ha
  apply ha.2.trans (hR1 (c a))
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
A : ∀ (a : ι), a ∈ t' → r a ≤ 1
A' : ∀ (a : ι), a ∈ t' → Set.Nonempty (B a)
⊢ ∃ u x, PairwiseDisjoint u B ∧ ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
A : ∀ (a : ι), a ∈ t' → r a ≤ 1
⊢ ∃ u x, PairwiseDisjoint u B ∧ ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
have A' : ∀ a ∈ t', (B a).Nonempty :=
  fun a hat' => Set.Nonempty.mono interior_subset (ht a hat'.1)
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
A : ∀ (a : ι), a ∈ t' → r a ≤ 1
A' : ∀ (a : ι), a ∈ t' → Set.Nonempty (B a)
a : ι
ha : a ∈ t'
⊢ 0 ≤ r a
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
A : ∀ (a : ι), a ∈ t' → r a ≤ 1
A' : ∀ (a : ι), a ∈ t' → Set.Nonempty (B a)
⊢ ∃ u x, PairwiseDisjoint u B ∧ ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
refine' exists_disjoint_subfamily_covering_enlargment B t' r 2 one_lt_two (fun a ha => _) 1 A A'
no goals
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
A : ∀ (a : ι), a ∈ t' → r a ≤ 1
A' : ∀ (a : ι), a ∈ t' → Set.Nonempty (B a)
a : ι
ha : a ∈ t'
⊢ 0 ≤ r a
exact nonempty_closedBall.1 ((A' a ha).mono (hB a ha.1))
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
a : ι
ha : a ∈ t'
⊢ r a ≤ 1
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
⊢ ∀ (a : ι), a ∈ t' → r a ≤ 1
intro a ha
no goals
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
a : ι
ha : a ∈ t'
⊢ r a ≤ 1
apply ha.2.trans (hR1 (c a))
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
⊢ ∃ K, ↑↑μ (closedBall x K) < ⊤ ∧ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
⊢ ∃ K, ↑↑μ (closedBall x K) < ⊤ ∧ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
have Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x := by
  intro a hav
  apply dist_le_add_of_nonempty_closedBall_inter_closedBall
  refine' hav.2.mono _
  apply inter_subset_inter _ ball_subset_closedBall
  exact hB a (ut (vu hav))
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
⊢ ∃ K, ↑↑μ (closedBall x K) < ⊤ ∧ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
⊢ ∃ K, ↑↑μ (closedBall x K) < ⊤ ∧ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
set R0 := sSup (r '' v) with R0_def
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
⊢ ∃ K, ↑↑μ (closedBall x K) < ⊤ ∧ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
⊢ ∃ K, ↑↑μ (closedBall x K) < ⊤ ∧ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
have R0_bdd : BddAbove (r '' v) := by
  refine' ⟨1, fun r' hr' => _⟩
  rcases (mem_image _ _ _).1 hr' with ⟨b, hb, rfl⟩
  exact le_trans (ut' (vu hb)).2 (hR1 (c b))
case inl
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R0 ≤ R x
⊢ ∃ K, ↑↑μ (closedBall x K) < ⊤ ∧ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K

case inr
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
⊢ ∃ K, ↑↑μ (closedBall x K) < ⊤ ∧ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
⊢ ∃ K, ↑↑μ (closedBall x K) < ⊤ ∧ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
rcases le_total R0 (R x) with (H | H)
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
a : ι
hav : a ∈ v
⊢ dist (c a) x ≤ r a + R x
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
⊢ ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
intro a hav
case h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
a : ι
hav : a ∈ v
⊢ Set.Nonempty (closedBall (c a) (r a) ∩ closedBall x (R x))
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
a : ι
hav : a ∈ v
⊢ dist (c a) x ≤ r a + R x
apply dist_le_add_of_nonempty_closedBall_inter_closedBall
case h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
a : ι
hav : a ∈ v
⊢ B a ∩ ball x (R x) ⊆ closedBall (c a) (r a) ∩ closedBall x (R x)
case h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
a : ι
hav : a ∈ v
⊢ Set.Nonempty (closedBall (c a) (r a) ∩ closedBall x (R x))
refine' hav.2.mono _
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
a : ι
hav : a ∈ v
⊢ B a ⊆ closedBall (c a) (r a)
case h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
a : ι
hav : a ∈ v
⊢ B a ∩ ball x (R x) ⊆ closedBall (c a) (r a) ∩ closedBall x (R x)
apply inter_subset_inter _ ball_subset_closedBall
no goals
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
a : ι
hav : a ∈ v
⊢ B a ⊆ closedBall (c a) (r a)
exact hB a (ut (vu hav))
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
r' : ℝ
hr' : r' ∈ r '' v
⊢ r' ≤ 1
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
⊢ BddAbove (r '' v)
refine' ⟨1, fun r' hr' => _⟩
case intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
b : ι
hb : b ∈ v
hr' : r b ∈ r '' v
⊢ r b ≤ 1
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
r' : ℝ
hr' : r' ∈ r '' v
⊢ r' ≤ 1
rcases (mem_image _ _ _).1 hr' with ⟨b, hb, rfl⟩
no goals
case intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
b : ι
hb : b ∈ v
hr' : r b ∈ r '' v
⊢ r b ≤ 1
exact le_trans (ut' (vu hb)).2 (hR1 (c b))
case inl
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R0 ≤ R x
a : ι
au : a ∈ u
hax : Set.Nonempty (B a ∩ ball x (R x))
⊢ B a ⊆ closedBall x (20 * R x)
case inl
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R0 ≤ R x
⊢ ∃ K, ↑↑μ (closedBall x K) < ⊤ ∧ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
refine' ⟨20 * R x, hRμ x, fun a au hax => _⟩
case inl
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R0 ≤ R x
a : ι
au : a ∈ u
hax : Set.Nonempty (B a ∩ ball x (R x))
⊢ closedBall (c a) (r a) ⊆ closedBall x (20 * R x)
case inl
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R0 ≤ R x
a : ι
au : a ∈ u
hax : Set.Nonempty (B a ∩ ball x (R x))
⊢ B a ⊆ closedBall x (20 * R x)
refine' (hB a (ut au)).trans _
case inl.h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R0 ≤ R x
a : ι
au : a ∈ u
hax : Set.Nonempty (B a ∩ ball x (R x))
⊢ r a + dist (c a) x ≤ 20 * R x
case inl
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R0 ≤ R x
a : ι
au : a ∈ u
hax : Set.Nonempty (B a ∩ ball x (R x))
⊢ closedBall (c a) (r a) ⊆ closedBall x (20 * R x)
apply closedBall_subset_closedBall'
case inl.h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R0 ≤ R x
a : ι
au : a ∈ u
hax : Set.Nonempty (B a ∩ ball x (R x))
this : r a ≤ R0
⊢ r a + dist (c a) x ≤ 20 * R x
case inl.h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R0 ≤ R x
a : ι
au : a ∈ u
hax : Set.Nonempty (B a ∩ ball x (R x))
⊢ r a + dist (c a) x ≤ 20 * R x
have : r a ≤ R0 := le_csSup R0_bdd (mem_image_of_mem _ ⟨au, hax⟩)
no goals
case inl.h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R0 ≤ R x
a : ι
au : a ∈ u
hax : Set.Nonempty (B a ∩ ball x (R x))
this : r a ≤ R0
⊢ r a + dist (c a) x ≤ 20 * R x
linarith [Idist_v a ⟨au, hax⟩, hR0 x]
case inr
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
⊢ ∃ K, ↑↑μ (closedBall x K) < ⊤ ∧ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
case inr
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
⊢ ∃ K, ↑↑μ (closedBall x K) < ⊤ ∧ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
have R0pos : 0 < R0 := (hR0 x).trans_le H
case inr
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
⊢ ∃ K, ↑↑μ (closedBall x K) < ⊤ ∧ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
case inr
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
⊢ ∃ K, ↑↑μ (closedBall x K) < ⊤ ∧ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
have vnonempty : v.Nonempty := by
  by_contra h
  rw [nonempty_iff_ne_empty, Classical.not_not] at h
  rw [h, image_empty, Real.sSup_empty] at R0_def
  exact lt_irrefl _ (R0pos.trans_le (le_of_eq R0_def))
case inr.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
⊢ ∃ K, ↑↑μ (closedBall x K) < ⊤ ∧ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
case inr
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
⊢ ∃ K, ↑↑μ (closedBall x K) < ⊤ ∧ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
obtain ⟨a, hav, R0a⟩ : ∃ a ∈ v, R0 / 2 < r a := by
  obtain ⟨r', r'mem, hr'⟩ : ∃ r' ∈ r '' v, R0 / 2 < r' :=
    exists_lt_of_lt_csSup (nonempty_image_iff.2 vnonempty) (half_lt_self R0pos)
  rcases (mem_image _ _ _).1 r'mem with ⟨a, hav, rfl⟩
  exact ⟨a, hav, hr'⟩
case inr.intro.intro.refine'_1
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
⊢ ↑↑μ (closedBall x (8 * R0)) < ⊤

case inr.intro.intro.refine'_2
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
⊢ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x (8 * R0)
case inr.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
⊢ ∃ K, ↑↑μ (closedBall x K) < ⊤ ∧ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
refine' ⟨8 * R0, _, _⟩
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
h : ¬Set.Nonempty v
⊢ False
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
⊢ Set.Nonempty v
by_contra h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
h : v = ∅
⊢ False
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
h : ¬Set.Nonempty v
⊢ False
rw [nonempty_iff_ne_empty, Classical.not_not] at h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = 0
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
h : v = ∅
⊢ False
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
h : v = ∅
⊢ False
rw [h, image_empty, Real.sSup_empty] at R0_def
no goals
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = 0
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
h : v = ∅
⊢ False
exact lt_irrefl _ (R0pos.trans_le (le_of_eq R0_def))
case intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
r' : ℝ
r'mem : r' ∈ r '' v
hr' : R0 / 2 < r'
⊢ ∃ a, a ∈ v ∧ R0 / 2 < r a
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
⊢ ∃ a, a ∈ v ∧ R0 / 2 < r a
obtain ⟨r', r'mem, hr'⟩ : ∃ r' ∈ r '' v, R0 / 2 < r' :=
  exists_lt_of_lt_csSup (nonempty_image_iff.2 vnonempty) (half_lt_self R0pos)
case intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
r'mem : r a ∈ r '' v
hr' : R0 / 2 < r a
⊢ ∃ a, a ∈ v ∧ R0 / 2 < r a
case intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
r' : ℝ
r'mem : r' ∈ r '' v
hr' : R0 / 2 < r'
⊢ ∃ a, a ∈ v ∧ R0 / 2 < r a
rcases (mem_image _ _ _).1 r'mem with ⟨a, hav, rfl⟩
no goals
case intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
r'mem : r a ∈ r '' v
hr' : R0 / 2 < r a
⊢ ∃ a, a ∈ v ∧ R0 / 2 < r a
exact ⟨a, hav, hr'⟩
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
⊢ closedBall x (8 * R0) ⊆ closedBall (c a) (20 * R (c a))
case inr.intro.intro.refine'_1
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
⊢ ↑↑μ (closedBall x (8 * R0)) < ⊤
apply lt_of_le_of_lt (measure_mono _) (hRμ (c a))
case h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
⊢ 8 * R0 + dist x (c a) ≤ 20 * R (c a)
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
⊢ closedBall x (8 * R0) ⊆ closedBall (c a) (20 * R (c a))
apply closedBall_subset_closedBall'
case h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
⊢ 8 * R0 + dist (c a) x ≤ 20 * R (c a)
case h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
⊢ 8 * R0 + dist x (c a) ≤ 20 * R (c a)
rw [dist_comm]
no goals
case h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
⊢ 8 * R0 + dist (c a) x ≤ 20 * R (c a)
linarith [Idist_v a hav, (ut' (vu hav)).2]
case inr.intro.intro.refine'_2
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
b : ι
bu : b ∈ u
hbx : Set.Nonempty (B b ∩ ball x (R x))
⊢ B b ⊆ closedBall x (8 * R0)
case inr.intro.intro.refine'_2
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
⊢ ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x (8 * R0)
intro b bu hbx
case inr.intro.intro.refine'_2
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
b : ι
bu : b ∈ u
hbx : Set.Nonempty (B b ∩ ball x (R x))
⊢ closedBall (c b) (r b) ⊆ closedBall x (8 * R0)
case inr.intro.intro.refine'_2
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
b : ι
bu : b ∈ u
hbx : Set.Nonempty (B b ∩ ball x (R x))
⊢ B b ⊆ closedBall x (8 * R0)
refine' (hB b (ut bu)).trans _
case inr.intro.intro.refine'_2.h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
b : ι
bu : b ∈ u
hbx : Set.Nonempty (B b ∩ ball x (R x))
⊢ r b + dist (c b) x ≤ 8 * R0
case inr.intro.intro.refine'_2
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
b : ι
bu : b ∈ u
hbx : Set.Nonempty (B b ∩ ball x (R x))
⊢ closedBall (c b) (r b) ⊆ closedBall x (8 * R0)
apply closedBall_subset_closedBall'
case inr.intro.intro.refine'_2.h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
b : ι
bu : b ∈ u
hbx : Set.Nonempty (B b ∩ ball x (R x))
this : r b ≤ R0
⊢ r b + dist (c b) x ≤ 8 * R0
case inr.intro.intro.refine'_2.h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
b : ι
bu : b ∈ u
hbx : Set.Nonempty (B b ∩ ball x (R x))
⊢ r b + dist (c b) x ≤ 8 * R0
have : r b ≤ R0 := le_csSup R0_bdd (mem_image_of_mem _ ⟨bu, hbx⟩)
no goals
case inr.intro.intro.refine'_2.h
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
Idist_v : ∀ (a : ι), a ∈ v → dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
R0pos : 0 < R0
vnonempty : Set.Nonempty v
a : ι
hav : a ∈ v
R0a : R0 / 2 < r a
b : ι
bu : b ∈ u
hbx : Set.Nonempty (B b ∩ ball x (R x))
this : r b ≤ R0
⊢ r b + dist (c b) x ≤ 8 * R0
linarith [Idist_v b ⟨bu, hbx⟩]
no goals
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
⊢ (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
calc
  (∑' a : v, μ (B a)) = μ (⋃ a ∈ v, B a) := by
    rw [measure_biUnion (u_count.mono vu) _ fun a ha => (h't _ (vu.trans ut ha)).measurableSet]
    exact u_disj.subset vu
  _ ≤ μ (closedBall x K) := (measure_mono (iUnion₂_subset fun a ha => hK a (vu ha) ha.2))
  _ < ∞ := μK
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
⊢ PairwiseDisjoint v fun a => B a
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
⊢ (∑' (a : ↑v), ↑↑μ (B ↑a)) = ↑↑μ (⋃ (a : ι) (_ : a ∈ v), B a)
rw [measure_biUnion (u_count.mono vu) _ fun a ha => (h't _ (vu.trans ut ha)).measurableSet]
no goals
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
⊢ PairwiseDisjoint v fun a => B a
exact u_disj.subset vu
no goals
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
⊢ 0 < ε / ↑C
simp only [ENNReal.div_pos_iff, εpos.ne', ENNReal.coe_ne_top, Ne.def, not_false_iff,
  and_self_iff]
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
⊢ z ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
⊢ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x) ⊆ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
intro z hz
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
⊢ z ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
⊢ z ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
set k := ⋃ (a : v) (ha : a ∈ w), B a
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
⊢ z ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
⊢ z ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
have k_closed : IsClosed k := isClosed_biUnion w.finite_toSet fun i _ => h't _ (ut (vu i.2))
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
z_notmem_k : ¬z ∈ k
⊢ z ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
⊢ z ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
have z_notmem_k : z ∉ k := by
  simp only [not_exists, exists_prop, mem_iUnion, mem_sep_iff, forall_exists_index,
    SetCoe.exists, not_and, exists_and_right, Subtype.coe_mk]
  intro b hbv _ h'z
  have : z ∈ (s \ ⋃ a ∈ u, B a) ∩ ⋃ a ∈ u, B a :=
    mem_inter (mem_of_mem_inter_left hz) (mem_biUnion (vu hbv) h'z)
  simpa only [diff_inter_self]
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
z_notmem_k : ¬z ∈ k
this : ball x (R x) \ k ∈ 𝓝 z
⊢ z ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
z_notmem_k : ¬z ∈ k
⊢ z ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
have : ball x (R x) \ k ∈ 𝓝 z := by
  apply IsOpen.mem_nhds (isOpen_ball.sdiff k_closed) _
  exact (mem_diff _).2 ⟨mem_of_mem_inter_right hz, z_notmem_k⟩
case intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
z_notmem_k : ¬z ∈ k
this : ball x (R x) \ k ∈ 𝓝 z
d : ℝ
dpos : 0 < d
hd : closedBall z d ⊆ ball x (R x) \ k
⊢ z ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
z_notmem_k : ¬z ∈ k
this : ball x (R x) \ k ∈ 𝓝 z
⊢ z ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
obtain ⟨d, dpos, hd⟩ : ∃ d, 0 < d ∧ closedBall z d ⊆ ball x (R x) \ k :=
  nhds_basis_closedBall.mem_iff.1 this
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
z_notmem_k : ¬z ∈ k
this : ball x (R x) \ k ∈ 𝓝 z
d : ℝ
dpos : 0 < d
hd : closedBall z d ⊆ ball x (R x) \ k
⊢ ∃ a, a ∈ t ∧ r a ≤ min d (R z) ∧ c a = z

case intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
case intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
z_notmem_k : ¬z ∈ k
this : ball x (R x) \ k ∈ 𝓝 z
d : ℝ
dpos : 0 < d
hd : closedBall z d ⊆ ball x (R x) \ k
⊢ z ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
obtain ⟨a, hat, ad, rfl⟩ : ∃ a ∈ t, r a ≤ min d (R z) ∧ c a = z
case intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
z_notmem_k : ¬z ∈ k
this : ball x (R x) \ k ∈ 𝓝 z
d : ℝ
dpos : 0 < d
hd : closedBall z d ⊆ ball x (R x) \ k
⊢ ∃ a, a ∈ t ∧ r a ≤ min d (R z) ∧ c a = z

case intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
exact hf z ((mem_diff _).1 (mem_of_mem_inter_left hz)).1 (min d (R z)) (lt_min dpos (hR0 z))
case intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
case intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
have ax : B a ⊆ ball x (R x) := by
  refine' (hB a hat).trans _
  refine' Subset.trans _ (hd.trans (diff_subset (ball x (R x)) k))
  exact closedBall_subset_closedBall (ad.trans (min_le_left _ _))
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
⊢ ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b

case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
case intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
obtain ⟨b, bu, ab, bdiam⟩ : ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
⊢ ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b

case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
exact hu a ⟨hat, ad.trans (min_le_right _ _)⟩
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
have bv : b ∈ v := by
  refine' ⟨bu, ab.mono _⟩
  rw [inter_comm]
  exact inter_subset_inter_right _ ax
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
let b' : v := ⟨b, bv⟩
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
have b'_notmem_w : b' ∉ w := by
  intro b'w
  have b'k : B b' ⊆ k := @Finset.subset_set_biUnion_of_mem _ _ _ (fun y : v => B y) _ b'w
  have : (ball x (R x) \ k ∩ k).Nonempty := by
    apply ab.mono (inter_subset_inter _ b'k)
    refine' ((hB _ hat).trans _).trans hd
    exact closedBall_subset_closedBall (ad.trans (min_le_left _ _))
  simpa only [diff_inter_self, Set.not_nonempty_empty]
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
let b'' : { a // a ∉ w } := ⟨b', b'_notmem_w⟩
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
zb : c a ∈ closedBall (c b) (3 * r b)
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
have zb : c a ∈ closedBall (c b) (3 * r b) := by
  rcases ab with ⟨e, ⟨ea, eb⟩⟩
  have A : dist (c a) e ≤ r a := mem_closedBall'.1 (hB a hat ea)
  have B : dist e (c b) ≤ r b := mem_closedBall.1 (hB b (ut bu) eb)
  simp only [mem_closedBall]
  linarith only [dist_triangle (c a) e (c b), A, B, bdiam]
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
zb : c a ∈ closedBall (c b) (3 * r b)
H : closedBall (c ↑↑b'') (3 * r ↑↑b'') ⊆ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)

case H
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
zb : c a ∈ closedBall (c b) (3 * r b)
⊢ closedBall (c ↑↑b'') (3 * r ↑↑b'') ⊆ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
zb : c a ∈ closedBall (c b) (3 * r b)
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
suffices H : closedBall (c b'') (3 * r b'') ⊆ ⋃ a : { a // a ∉ w }, closedBall (c a) (3 * r a)
case H
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
zb : c a ∈ closedBall (c b) (3 * r b)
⊢ closedBall (c ↑↑b'') (3 * r ↑↑b'') ⊆ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
case intro.intro.intro.intro.intro.intro.intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
zb : c a ∈ closedBall (c b) (3 * r b)
H : closedBall (c ↑↑b'') (3 * r ↑↑b'') ⊆ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
⊢ c a ∈ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)

case H
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
zb : c a ∈ closedBall (c b) (3 * r b)
⊢ closedBall (c ↑↑b'') (3 * r ↑↑b'') ⊆ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
exact H zb
no goals
case H
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
zb : c a ∈ closedBall (c b) (3 * r b)
⊢ closedBall (c ↑↑b'') (3 * r ↑↑b'') ⊆ ⋃ (a : { a // ¬a ∈ w }), closedBall (c ↑↑a) (3 * r ↑↑a)
exact subset_iUnion (fun a : { a // a ∉ w } => closedBall (c a) (3 * r a)) b''
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
⊢ ∀ (x_1 : ι) (x_2 : x_1 ∈ u ∧ Set.Nonempty (B x_1 ∩ ball x (R x))),
    { val := x_1, property := (_ : x_1 ∈ {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}) } ∈ w → ¬z ∈ B x_1
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
⊢ ¬z ∈ k
simp only [not_exists, exists_prop, mem_iUnion, mem_sep_iff, forall_exists_index,
  SetCoe.exists, not_and, exists_and_right, Subtype.coe_mk]
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
b : ι
hbv : b ∈ u ∧ Set.Nonempty (B b ∩ ball x (R x))
h✝ : { val := b, property := (_ : b ∈ {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}) } ∈ w
h'z : z ∈ B b
⊢ False
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
⊢ ∀ (x_1 : ι) (x_2 : x_1 ∈ u ∧ Set.Nonempty (B x_1 ∩ ball x (R x))),
    { val := x_1, property := (_ : x_1 ∈ {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}) } ∈ w → ¬z ∈ B x_1
intro b hbv _ h'z
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
b : ι
hbv : b ∈ u ∧ Set.Nonempty (B b ∩ ball x (R x))
h✝ : { val := b, property := (_ : b ∈ {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}) } ∈ w
h'z : z ∈ B b
this : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ⋃ (a : ι) (_ : a ∈ u), B a
⊢ False
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
b : ι
hbv : b ∈ u ∧ Set.Nonempty (B b ∩ ball x (R x))
h✝ : { val := b, property := (_ : b ∈ {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}) } ∈ w
h'z : z ∈ B b
⊢ False
have : z ∈ (s \ ⋃ a ∈ u, B a) ∩ ⋃ a ∈ u, B a :=
  mem_inter (mem_of_mem_inter_left hz) (mem_biUnion (vu hbv) h'z)
no goals
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
b : ι
hbv : b ∈ u ∧ Set.Nonempty (B b ∩ ball x (R x))
h✝ : { val := b, property := (_ : b ∈ {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}) } ∈ w
h'z : z ∈ B b
this : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ⋃ (a : ι) (_ : a ∈ u), B a
⊢ False
simpa only [diff_inter_self]
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
z_notmem_k : ¬z ∈ k
⊢ z ∈ ball x (R x) \ k
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
z_notmem_k : ¬z ∈ k
⊢ ball x (R x) \ k ∈ 𝓝 z
apply IsOpen.mem_nhds (isOpen_ball.sdiff k_closed) _
no goals
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
z : α
hz : z ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
z_notmem_k : ¬z ∈ k
⊢ z ∈ ball x (R x) \ k
exact (mem_diff _).2 ⟨mem_of_mem_inter_right hz, z_notmem_k⟩
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
⊢ closedBall (c a) (r a) ⊆ ball x (R x)
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
⊢ B a ⊆ ball x (R x)
refine' (hB a hat).trans _
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
⊢ closedBall (c a) (r a) ⊆ closedBall (c a) d
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
⊢ closedBall (c a) (r a) ⊆ ball x (R x)
refine' Subset.trans _ (hd.trans (diff_subset (ball x (R x)) k))
no goals
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
⊢ closedBall (c a) (r a) ⊆ closedBall (c a) d
exact closedBall_subset_closedBall (ad.trans (min_le_left _ _))
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
⊢ B a ∩ B b ⊆ B b ∩ ball x (R x)
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
⊢ b ∈ v
refine' ⟨bu, ab.mono _⟩
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
⊢ B b ∩ B a ⊆ B b ∩ ball x (R x)
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
⊢ B a ∩ B b ⊆ B b ∩ ball x (R x)
rw [inter_comm]
no goals
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
⊢ B b ∩ B a ⊆ B b ∩ ball x (R x)
exact inter_subset_inter_right _ ax
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'w : b' ∈ w
⊢ False
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
⊢ ¬b' ∈ w
intro b'w
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'w : b' ∈ w
b'k : B ↑b' ⊆ k
⊢ False
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'w : b' ∈ w
⊢ False
have b'k : B b' ⊆ k := @Finset.subset_set_biUnion_of_mem _ _ _ (fun y : v => B y) _ b'w
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this✝ : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'w : b' ∈ w
b'k : B ↑b' ⊆ k
this : Set.Nonempty (ball x (R x) \ k ∩ k)
⊢ False
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'w : b' ∈ w
b'k : B ↑b' ⊆ k
⊢ False
have : (ball x (R x) \ k ∩ k).Nonempty := by
  apply ab.mono (inter_subset_inter _ b'k)
  refine' ((hB _ hat).trans _).trans hd
  exact closedBall_subset_closedBall (ad.trans (min_le_left _ _))
no goals
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this✝ : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'w : b' ∈ w
b'k : B ↑b' ⊆ k
this : Set.Nonempty (ball x (R x) \ k ∩ k)
⊢ False
simpa only [diff_inter_self, Set.not_nonempty_empty]
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'w : b' ∈ w
b'k : B ↑b' ⊆ k
⊢ B a ⊆ ball x (R x) \ k
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'w : b' ∈ w
b'k : B ↑b' ⊆ k
⊢ Set.Nonempty (ball x (R x) \ k ∩ k)
apply ab.mono (inter_subset_inter _ b'k)
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'w : b' ∈ w
b'k : B ↑b' ⊆ k
⊢ closedBall (c a) (r a) ⊆ closedBall (c a) d
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'w : b' ∈ w
b'k : B ↑b' ⊆ k
⊢ B a ⊆ ball x (R x) \ k
refine' ((hB _ hat).trans _).trans hd
no goals
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'w : b' ∈ w
b'k : B ↑b' ⊆ k
⊢ closedBall (c a) (r a) ⊆ closedBall (c a) d
exact closedBall_subset_closedBall (ad.trans (min_le_left _ _))
case intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
e : α
ea : e ∈ B a
eb : e ∈ B b
⊢ c a ∈ closedBall (c b) (3 * r b)
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
ab : Set.Nonempty (B a ∩ B b)
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
⊢ c a ∈ closedBall (c b) (3 * r b)
rcases ab with ⟨e, ⟨ea, eb⟩⟩
case intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
e : α
ea : e ∈ B a
eb : e ∈ B b
A : dist (c a) e ≤ r a
⊢ c a ∈ closedBall (c b) (3 * r b)
case intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
e : α
ea : e ∈ B a
eb : e ∈ B b
⊢ c a ∈ closedBall (c b) (3 * r b)
have A : dist (c a) e ≤ r a := mem_closedBall'.1 (hB a hat ea)
case intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B✝ : ι → Set α
hB : ∀ (a : ι), a ∈ t → B✝ a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B✝ a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B✝ a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B✝ a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B✝
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B✝ a ∩ B✝ b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B✝ a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B✝ a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B✝ a ∩ ball x (R x)) → B✝ a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B✝ ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B✝ ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B✝ ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B✝ a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B✝ a ⊆ ball x (R x)
b : ι
bu : b ∈ u
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
e : α
ea : e ∈ B✝ a
eb : e ∈ B✝ b
A : dist (c a) e ≤ r a
B : dist e (c b) ≤ r b
⊢ c a ∈ closedBall (c b) (3 * r b)
case intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ (a : ι), a ∈ t → B a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B a ∩ B b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B a ∩ ball x (R x)) → B a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B a ⊆ ball x (R x)
b : ι
bu : b ∈ u
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
e : α
ea : e ∈ B a
eb : e ∈ B b
A : dist (c a) e ≤ r a
⊢ c a ∈ closedBall (c b) (3 * r b)
have B : dist e (c b) ≤ r b := mem_closedBall.1 (hB b (ut bu) eb)
case intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B✝ : ι → Set α
hB : ∀ (a : ι), a ∈ t → B✝ a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B✝ a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B✝ a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B✝ a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B✝
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B✝ a ∩ B✝ b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B✝ a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B✝ a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B✝ a ∩ ball x (R x)) → B✝ a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B✝ ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B✝ ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B✝ ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B✝ a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B✝ a ⊆ ball x (R x)
b : ι
bu : b ∈ u
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
e : α
ea : e ∈ B✝ a
eb : e ∈ B✝ b
A : dist (c a) e ≤ r a
B : dist e (c b) ≤ r b
⊢ dist (c a) (c b) ≤ 3 * r b
case intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B✝ : ι → Set α
hB : ∀ (a : ι), a ∈ t → B✝ a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B✝ a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B✝ a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B✝ a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B✝
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B✝ a ∩ B✝ b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B✝ a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B✝ a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B✝ a ∩ ball x (R x)) → B✝ a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B✝ ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B✝ ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B✝ ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B✝ a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B✝ a ⊆ ball x (R x)
b : ι
bu : b ∈ u
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
e : α
ea : e ∈ B✝ a
eb : e ∈ B✝ b
A : dist (c a) e ≤ r a
B : dist e (c b) ≤ r b
⊢ c a ∈ closedBall (c b) (3 * r b)
simp only [mem_closedBall]
no goals
case intro.intro
α : Type u_1
ι : Type u_2
inst✝⁴ : MetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : MeasureTheory.Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B✝ : ι → Set α
hB : ∀ (a : ι), a ∈ t → B✝ a ⊆ closedBall (c a) (r a)
μB : ∀ (a : ι), a ∈ t → ↑↑μ (closedBall (c a) (3 * r a)) ≤ ↑C * ↑↑μ (B✝ a)
ht : ∀ (a : ι), a ∈ t → Set.Nonempty (interior (B✝ a))
h't : ∀ (a : ι), a ∈ t → IsClosed (B✝ a)
hf : ∀ (x : α), x ∈ s → ∀ (ε : ℝ), ε > 0 → ∃ a, a ∈ t ∧ r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), ↑↑μ (closedBall x (20 * R x)) < ⊤
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : PairwiseDisjoint u B✝
hu : ∀ (a : ι), a ∈ t' → ∃ b, b ∈ u ∧ Set.Nonempty (B✝ a ∩ B✝ b) ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : Set.Countable u
x : α
x✝ : x ∈ s \ ⋃ (a : ι) (_ : a ∈ u), B✝ a
v : Set ι := {a | a ∈ u ∧ Set.Nonempty (B✝ a ∩ ball x (R x))}
vu : v ⊆ u
K : ℝ
μK : ↑↑μ (closedBall x K) < ⊤
hK : ∀ (a : ι), a ∈ u → Set.Nonempty (B✝ a ∩ ball x (R x)) → B✝ a ⊆ closedBall x K
ε : ℝ≥0∞
εpos : 0 < ε
I : (∑' (a : ↑v), ↑↑μ (B✝ ↑a)) < ⊤
w : Finset ↑v
hw : (∑' (a : { a // ¬a ∈ w }), ↑↑μ (B✝ ↑↑a)) < ε / ↑C
k : Set α := ⋃ (a : ↑v) (_ : a ∈ w), B✝ ↑a
k_closed : IsClosed k
d : ℝ
dpos : 0 < d
a : ι
hat : a ∈ t
hz : c a ∈ (s \ ⋃ (a : ι) (_ : a ∈ u), B✝ a) ∩ ball x (R x)
z_notmem_k : ¬c a ∈ k
this : ball x (R x) \ k ∈ 𝓝 (c a)
hd : closedBall (c a) d ⊆ ball x (R x) \ k
ad : r a ≤ min d (R (c a))
ax : B✝ a ⊆ ball x (R x)
b : ι
bu : b ∈ u
bdiam : r a ≤ 2 * r b
bv : b ∈ v
b' : ↑v := { val := b, property := bv }
b'_notmem_w : ¬b' ∈ w
b'' : { a // ¬a ∈ w } := { val := b', property := b'_notmem_w }
e : α
ea : e ∈ B✝ a
eb : e ∈ B✝ b
A : dist (c a) e ≤ r a
B : dist e (c b) ≤ r b
⊢ dist (c a) (c b) ≤ 3 * r b
linarith only [dist_triangle (c a) e (c b), A, B, bdiam]