α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > ↑↑μ A
⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑μ U < r
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
⊢ OuterRegular μ
refine' ⟨fun A hA r hr => _⟩
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > ↑↑μ A
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑μ U < r
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > ↑↑μ A
⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑μ U < r
have hm : ∀ n, MeasurableSet (s.set n) := fun n => (s.set_mem n).1.measurableSet
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > ↑↑μ A
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑μ U < r
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > ↑↑μ A
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑μ U < r
haveI : ∀ n, OuterRegular (μ.restrict (s.set n)) := fun n => (s.set_mem n).2
case intro.intro.intro.intro
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
⊢ ∃ U, (U ⊇ ⋃ (n : ℕ), A n) ∧ IsOpen U ∧ ↑↑μ U < r
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > ↑↑μ A
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
⊢ ∃ U, U ⊇ A ∧ IsOpen U ∧ ↑↑μ U < r
obtain ⟨A, hAm, hAs, hAd, rfl⟩ :
  ∃ A' : ℕ → Set α,
    (∀ n, MeasurableSet (A' n)) ∧
      (∀ n, A' n ⊆ s.set n) ∧ Pairwise (Disjoint on A') ∧ A = ⋃ n, A' n := by
  refine'
    ⟨fun n => A ∩ disjointed s.set n, fun n => hA.inter (MeasurableSet.disjointed hm _), fun n =>
      (inter_subset_right _ _).trans (disjointed_subset _ _),
      (disjoint_disjointed s.set).mono fun k l hkl => hkl.mono inf_le_right inf_le_right, _⟩
  rw [← inter_iUnion, iUnion_disjointed, s.spanning, inter_univ]
case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
⊢ ∃ U, (U ⊇ ⋃ (n : ℕ), A n) ∧ IsOpen U ∧ ↑↑μ U < r
case intro.intro.intro.intro
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
⊢ ∃ U, (U ⊇ ⋃ (n : ℕ), A n) ∧ IsOpen U ∧ ↑↑μ U < r
rcases ENNReal.exists_pos_sum_of_countable' (tsub_pos_iff_lt.2 hr).ne' ℕ with ⟨δ, δ0, hδε⟩
case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
⊢ ∃ U, (U ⊇ ⋃ (n : ℕ), A n) ∧ IsOpen U ∧ ↑↑μ U < r
case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
⊢ ∃ U, (U ⊇ ⋃ (n : ℕ), A n) ∧ IsOpen U ∧ ↑↑μ U < r
rw [lt_tsub_iff_right, add_comm] at hδε
case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this✝ : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
this : ∀ (n : ℕ), ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n
⊢ ∃ U, (U ⊇ ⋃ (n : ℕ), A n) ∧ IsOpen U ∧ ↑↑μ U < r
case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
⊢ ∃ U, (U ⊇ ⋃ (n : ℕ), A n) ∧ IsOpen U ∧ ↑↑μ U < r
have : ∀ n, ∃ (U : _)(_ : U ⊇ A n), IsOpen U ∧ μ U < μ (A n) + δ n := by
  intro n
  have H₁ : ∀ t, μ.restrict (s.set n) t = μ (t ∩ s.set n) := fun t => restrict_apply' (hm n)
  have Ht : μ.restrict (s.set n) (A n) ≠ ⊤ := by
    rw [H₁]
    exact ((measure_mono <| inter_subset_right _ _).trans_lt (s.finite n)).ne
  rcases(A n).exists_isOpen_lt_add Ht (δ0 n).ne' with ⟨U, hAU, hUo, hU⟩
  rw [H₁, H₁, inter_eq_self_of_subset_left (hAs _)] at hU
  exact ⟨U ∩ s.set n, subset_inter hAU (hAs _), hUo.inter (s.set_mem n).1, hU⟩
case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
U : ℕ → Set α
hAU : ∀ (n : ℕ), U n ⊇ A n
hUo : ∀ (n : ℕ), IsOpen (U n)
hU : ∀ (n : ℕ), ↑↑μ (U n) < ↑↑μ (A n) + δ n
⊢ ∃ U, (U ⊇ ⋃ (n : ℕ), A n) ∧ IsOpen U ∧ ↑↑μ U < r
case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this✝ : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
this : ∀ (n : ℕ), ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n
⊢ ∃ U, (U ⊇ ⋃ (n : ℕ), A n) ∧ IsOpen U ∧ ↑↑μ U < r
choose U hAU hUo hU using this
case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
U : ℕ → Set α
hAU : ∀ (n : ℕ), U n ⊇ A n
hUo : ∀ (n : ℕ), IsOpen (U n)
hU : ∀ (n : ℕ), ↑↑μ (U n) < ↑↑μ (A n) + δ n
⊢ ↑↑μ (⋃ (n : ℕ), U n) < r
case intro.intro.intro.intro.intro.intro
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
U : ℕ → Set α
hAU : ∀ (n : ℕ), U n ⊇ A n
hUo : ∀ (n : ℕ), IsOpen (U n)
hU : ∀ (n : ℕ), ↑↑μ (U n) < ↑↑μ (A n) + δ n
⊢ ∃ U, (U ⊇ ⋃ (n : ℕ), A n) ∧ IsOpen U ∧ ↑↑μ U < r
refine' ⟨⋃ n, U n, iUnion_mono hAU, isOpen_iUnion hUo, _⟩
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > ↑↑μ A
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
⊢ A = ⋃ (n : ℕ), (fun n => A ∩ disjointed s.set n) n
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > ↑↑μ A
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
⊢ ∃ A',
    (∀ (n : ℕ), MeasurableSet (A' n)) ∧
      (∀ (n : ℕ), A' n ⊆ FiniteSpanningSetsIn.set s n) ∧ Pairwise (Disjoint on A') ∧ A = ⋃ (n : ℕ), A' n
refine'
  ⟨fun n => A ∩ disjointed s.set n, fun n => hA.inter (MeasurableSet.disjointed hm _), fun n =>
    (inter_subset_right _ _).trans (disjointed_subset _ _),
    (disjoint_disjointed s.set).mono fun k l hkl => hkl.mono inf_le_right inf_le_right, _⟩
no goals
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > ↑↑μ A
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
⊢ A = ⋃ (n : ℕ), (fun n => A ∩ disjointed s.set n) n
rw [← inter_iUnion, iUnion_disjointed, s.spanning, inter_univ]
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
n : ℕ
⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
⊢ ∀ (n : ℕ), ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n
intro n
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
n : ℕ
H₁ : ∀ (t : Set α), ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) t = ↑↑μ (t ∩ FiniteSpanningSetsIn.set s n)
⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
n : ℕ
⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n
have H₁ : ∀ t, μ.restrict (s.set n) t = μ (t ∩ s.set n) := fun t => restrict_apply' (hm n)
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
n : ℕ
H₁ : ∀ (t : Set α), ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) t = ↑↑μ (t ∩ FiniteSpanningSetsIn.set s n)
Ht : ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) (A n) ≠ ⊤
⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
n : ℕ
H₁ : ∀ (t : Set α), ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) t = ↑↑μ (t ∩ FiniteSpanningSetsIn.set s n)
⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n
have Ht : μ.restrict (s.set n) (A n) ≠ ⊤ := by
  rw [H₁]
  exact ((measure_mono <| inter_subset_right _ _).trans_lt (s.finite n)).ne
case intro.intro.intro
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
n : ℕ
H₁ : ∀ (t : Set α), ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) t = ↑↑μ (t ∩ FiniteSpanningSetsIn.set s n)
Ht : ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) (A n) ≠ ⊤
U : Set α
hAU : U ⊇ A n
hUo : IsOpen U
hU : ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) U < ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) (A n) + δ n
⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
n : ℕ
H₁ : ∀ (t : Set α), ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) t = ↑↑μ (t ∩ FiniteSpanningSetsIn.set s n)
Ht : ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) (A n) ≠ ⊤
⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n
rcases(A n).exists_isOpen_lt_add Ht (δ0 n).ne' with ⟨U, hAU, hUo, hU⟩
case intro.intro.intro
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
n : ℕ
H₁ : ∀ (t : Set α), ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) t = ↑↑μ (t ∩ FiniteSpanningSetsIn.set s n)
Ht : ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) (A n) ≠ ⊤
U : Set α
hAU : U ⊇ A n
hUo : IsOpen U
hU : ↑↑μ (U ∩ FiniteSpanningSetsIn.set s n) < ↑↑μ (A n) + δ n
⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n
case intro.intro.intro
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
n : ℕ
H₁ : ∀ (t : Set α), ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) t = ↑↑μ (t ∩ FiniteSpanningSetsIn.set s n)
Ht : ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) (A n) ≠ ⊤
U : Set α
hAU : U ⊇ A n
hUo : IsOpen U
hU : ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) U < ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) (A n) + δ n
⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n
rw [H₁, H₁, inter_eq_self_of_subset_left (hAs _)] at hU
no goals
case intro.intro.intro
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
n : ℕ
H₁ : ∀ (t : Set α), ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) t = ↑↑μ (t ∩ FiniteSpanningSetsIn.set s n)
Ht : ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) (A n) ≠ ⊤
U : Set α
hAU : U ⊇ A n
hUo : IsOpen U
hU : ↑↑μ (U ∩ FiniteSpanningSetsIn.set s n) < ↑↑μ (A n) + δ n
⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ↑↑μ (A n) + δ n
exact ⟨U ∩ s.set n, subset_inter hAU (hAs _), hUo.inter (s.set_mem n).1, hU⟩
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
n : ℕ
H₁ : ∀ (t : Set α), ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) t = ↑↑μ (t ∩ FiniteSpanningSetsIn.set s n)
⊢ ↑↑μ (A n ∩ FiniteSpanningSetsIn.set s n) ≠ ⊤
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
n : ℕ
H₁ : ∀ (t : Set α), ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) t = ↑↑μ (t ∩ FiniteSpanningSetsIn.set s n)
⊢ ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) (A n) ≠ ⊤
rw [H₁]
no goals
α : Type u_1
β : Type ?u.18899
inst✝² : MeasurableSpace α
inst✝¹ : TopologicalSpace α
μ✝ : Measure α
inst✝ : OpensMeasurableSpace α
μ : Measure α
s : FiniteSpanningSetsIn μ {U | IsOpen U ∧ OuterRegular (restrict μ U)}
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (FiniteSpanningSetsIn.set s n)
this : ∀ (n : ℕ), OuterRegular (restrict μ (FiniteSpanningSetsIn.set s n))
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ FiniteSpanningSetsIn.set s n
hA : MeasurableSet (⋃ (n : ℕ), A n)
hr : r > ↑↑μ (⋃ (n : ℕ), A n)
hAd : Pairwise (Disjoint on A)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε✝ : (∑' (i : ℕ), δ i) < r - ↑↑μ (⋃ (n : ℕ), A n)
hδε : (↑↑μ (⋃ (n : ℕ), A n) + ∑' (i : ℕ), δ i) < r
n : ℕ
H₁ : ∀ (t : Set α), ↑↑(restrict μ (FiniteSpanningSetsIn.set s n)) t = ↑↑μ (t ∩ FiniteSpanningSetsIn.set s n)
⊢ ↑↑μ (A n ∩ FiniteSpanningSetsIn.set s n) ≠ ⊤
exact ((measure_mono <| inter_subset_right _ _).trans_lt (s.finite n)).ne