α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
let r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / ((i : ℝ) + 1)
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
have h₀ : ∀ i, p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i) := by
  rintro i ⟨-, hi⟩; congr! 1; change r i = ite (0 < r i) (r i) _; simp [hi]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
have h₁ : ∀ i, p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i) := by
  rintro i ⟨-, hi⟩; simp only [hi, mul_ite, if_true]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
have h₂ : ∀ i, p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i) := by
  rintro i ⟨-, hi⟩
  have hi' : M * r i ≤ 0 := mul_nonpos_of_nonneg_of_nonpos hM.le hi
  rw [cthickening_of_nonpos hi, cthickening_of_nonpos hi']
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
hp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
have hp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0 := by
  ext i; simp [← and_or_left, lt_or_le 0 (r i)]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
hp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
⊢ ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ 0 < r i) ⊔
      blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ r i ≤ 0) =ᵐ[μ]
    (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ 0 < r i) ⊔
      blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ r i ≤ 0
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
hp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
rw [hp, blimsup_or_eq_sup, blimsup_or_eq_sup]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
hp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
⊢ ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ 0 < r i) ∪
      blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ r i ≤ 0) =ᵐ[μ]
    (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ 0 < r i) ∪
      blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ r i ≤ 0
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
hp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
⊢ ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ 0 < r i) ⊔
      blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ r i ≤ 0) =ᵐ[μ]
    (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ 0 < r i) ⊔
      blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ r i ≤ 0
simp only [sup_eq_union]
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
hp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
⊢ ((blimsup (fun x => cthickening (M * r' x) (s x)) atTop fun x => p x ∧ 0 < r x) ∪
      blimsup (fun x => cthickening (r x) (s x)) atTop fun x => p x ∧ r x ≤ 0) =ᵐ[μ]
    (blimsup (fun x => cthickening (r' x) (s x)) atTop fun x => p x ∧ 0 < r x) ∪
      blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ r i ≤ 0
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
hp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
⊢ ((blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ 0 < r i) ∪
      blimsup (fun i => cthickening (M * r i) (s i)) atTop fun i => p i ∧ r i ≤ 0) =ᵐ[μ]
    (blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ 0 < r i) ∪
      blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ r i ≤ 0
rw [blimsup_congr (eventually_of_forall h₀), blimsup_congr (eventually_of_forall h₁),
  blimsup_congr (eventually_of_forall h₂)]
no goals
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
hp : p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
⊢ ((blimsup (fun x => cthickening (M * r' x) (s x)) atTop fun x => p x ∧ 0 < r x) ∪
      blimsup (fun x => cthickening (r x) (s x)) atTop fun x => p x ∧ r x ≤ 0) =ᵐ[μ]
    (blimsup (fun x => cthickening (r' x) (s x)) atTop fun x => p x ∧ 0 < r x) ∪
      blimsup (fun i => cthickening (r i) (s i)) atTop fun i => p i ∧ r i ≤ 0
exact ae_eq_set_union (this (fun i => p i ∧ 0 < r i) hr') (ae_eq_refl _)
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
⊢ ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
⊢ ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
clear p hr r
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
⊢ ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
intro p r hr
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
hr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
have hr' : Tendsto (fun i => M * r i) atTop (𝓝[>] 0) := by
  convert TendstoNhdsWithinIoi.const_mul hM hr <;> simp only [MulZeroClass.mul_zero]
case refine'_1
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
hr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p

case refine'_2
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
hr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)
⊢ blimsup (fun i => cthickening (r i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => cthickening (M * r i) (s i)) atTop p
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
hr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
refine' eventuallyLE_antisymm_iff.mpr ⟨_, _⟩
no goals
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
⊢ Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)
convert TendstoNhdsWithinIoi.const_mul hM hr <;> simp only [MulZeroClass.mul_zero]
no goals
case refine'_1
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
hr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)
⊢ blimsup (fun i => cthickening (M * r i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
exact blimsup_cthickening_ae_le_of_eventually_mul_le μ p (inv_pos.mpr hM) hr'
  (eventually_of_forall fun i => by rw [inv_mul_cancel_left₀ hM.ne' (r i)])
no goals
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
hr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)
i : ℕ
⊢ M⁻¹ * (M * r i) ≤ r i
rw [inv_mul_cancel_left₀ hM.ne' (r i)]
no goals
case refine'_2
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
s : ℕ → Set α
M : ℝ
hM : 0 < M
p : ℕ → Prop
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝[Ioi 0] 0)
hr' : Tendsto (fun i => M * r i) atTop (𝓝[Ioi 0] 0)
⊢ blimsup (fun i => cthickening (r i) (s i)) atTop p ≤ᵐ[μ] blimsup (fun i => cthickening (M * r i) (s i)) atTop p
exact blimsup_cthickening_ae_le_of_eventually_mul_le μ p hM hr
  (eventually_of_forall fun i => le_refl _)
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
i : ℕ
⊢ r' i ∈ Ioi 0
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
⊢ Tendsto r' atTop (𝓝[Ioi 0] 0)
refine' tendsto_nhdsWithin_iff.mpr
  ⟨Tendsto.if' hr tendsto_one_div_add_atTop_nhds_0_nat, eventually_of_forall fun i => _⟩
case pos
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
i : ℕ
hi : 0 < r i
⊢ r' i ∈ Ioi 0

case neg
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
i : ℕ
hi : ¬0 < r i
⊢ r' i ∈ Ioi 0
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
i : ℕ
⊢ r' i ∈ Ioi 0
by_cases hi : 0 < r i
no goals
case pos
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
i : ℕ
hi : 0 < r i
⊢ r' i ∈ Ioi 0
simp [hi]
case neg
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
i : ℕ
hi : ¬0 < r i
⊢ 0 < ↑i + 1
case neg
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
i : ℕ
hi : ¬0 < r i
⊢ r' i ∈ Ioi 0
simp only [hi, one_div, mem_Ioi, if_false, inv_pos]
no goals
case neg
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
i : ℕ
hi : ¬0 < r i
⊢ 0 < ↑i + 1
positivity
case intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
i : ℕ
hi : 0 < r i
⊢ cthickening (r i) (s i) = cthickening (r' i) (s i)
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
⊢ ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
rintro i ⟨-, hi⟩
case intro.h.e'_3
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
i : ℕ
hi : 0 < r i
⊢ r i = r' i
case intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
i : ℕ
hi : 0 < r i
⊢ cthickening (r i) (s i) = cthickening (r' i) (s i)
congr! 1
case intro.h.e'_3
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
i : ℕ
hi : 0 < r i
⊢ r i = if 0 < r i then r i else 1 / (↑i + 1)
case intro.h.e'_3
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
i : ℕ
hi : 0 < r i
⊢ r i = r' i
change r i = ite (0 < r i) (r i) _
no goals
case intro.h.e'_3
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
i : ℕ
hi : 0 < r i
⊢ r i = if 0 < r i then r i else 1 / (↑i + 1)
simp [hi]
case intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
i : ℕ
hi : 0 < r i
⊢ cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
⊢ ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
rintro i ⟨-, hi⟩
no goals
case intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
i : ℕ
hi : 0 < r i
⊢ cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
simp only [hi, mul_ite, if_true]
case intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
i : ℕ
hi : r i ≤ 0
⊢ cthickening (M * r i) (s i) = cthickening (r i) (s i)
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
⊢ ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
rintro i ⟨-, hi⟩
case intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
i : ℕ
hi : r i ≤ 0
hi' : M * r i ≤ 0
⊢ cthickening (M * r i) (s i) = cthickening (r i) (s i)
case intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
i : ℕ
hi : r i ≤ 0
⊢ cthickening (M * r i) (s i) = cthickening (r i) (s i)
have hi' : M * r i ≤ 0 := mul_nonpos_of_nonneg_of_nonpos hM.le hi
no goals
case intro
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
i : ℕ
hi : r i ≤ 0
hi' : M * r i ≤ 0
⊢ cthickening (M * r i) (s i) = cthickening (r i) (s i)
rw [cthickening_of_nonpos hi, cthickening_of_nonpos hi']
case h.a
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
i : ℕ
⊢ p i ↔ p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
⊢ p = fun i => p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
ext i
no goals
case h.a
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : MeasureTheory.Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r : ℕ → ℝ
hr : Tendsto r atTop (𝓝 0)
this :
  ∀ (p : ℕ → Prop) {r : ℕ → ℝ},
    Tendsto r atTop (𝓝[Ioi 0] 0) →
      blimsup (fun i => cthickening (M * r i) (s i)) atTop p =ᵐ[μ] blimsup (fun i => cthickening (r i) (s i)) atTop p
r' : ℕ → ℝ := fun i => if 0 < r i then r i else 1 / (↑i + 1)
hr' : Tendsto r' atTop (𝓝[Ioi 0] 0)
h₀ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (r i) (s i) = cthickening (r' i) (s i)
h₁ : ∀ (i : ℕ), p i ∧ 0 < r i → cthickening (M * r i) (s i) = cthickening (M * r' i) (s i)
h₂ : ∀ (i : ℕ), p i ∧ r i ≤ 0 → cthickening (M * r i) (s i) = cthickening (r i) (s i)
i : ℕ
⊢ p i ↔ p i ∧ 0 < r i ∨ p i ∧ r i ≤ 0
simp [← and_or_left, lt_or_le 0 (r i)]