Theorem: blimsup_cthickening_ae_le_of_eventually_mul_le
File: Mathlib/MeasureTheory/Covering/LiminfLimsup.lean

Proof steps:
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
⊢ blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᶠ[ae μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
state_after: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
⊢ blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᶠ[ae μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
tactic: let R₁ i := max 0 (r₁ i)
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
⊢ blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᶠ[ae μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
state_after: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
⊢ blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᶠ[ae μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
tactic: let R₂ i := max 0 (r₂ i)
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
⊢ blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᶠ[ae μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
state_after: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
⊢ blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᶠ[ae μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
tactic: have hRp : 0 ≤ R₁ := fun i => le_max_left 0 (r₁ i)
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
hMr : ∀ᶠ (i : ℕ) in atTop, M * r₁ i ≤ r₂ i
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
⊢ blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᶠ[ae μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
state_after: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
⊢ blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᶠ[ae μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
tactic: replace hMr : ∀ᶠ i in atTop, M * R₁ i ≤ R₂ i := by
    refine hMr.mono fun i hi ↦ ?_
    rw [mul_max_of_nonneg _ _ hM.le, mul_zero]
    exact max_le_max (le_refl 0) hi
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
⊢ blimsup (fun i => cthickening (r₁ i) (s i)) atTop p ≤ᶠ[ae μ] blimsup (fun i => cthickening (r₂ i) (s i)) atTop p
state_after: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
⊢ blimsup (fun i => cthickening (max 0 (r₁ i)) (s i)) atTop p ≤ᶠ[ae μ]
    blimsup (fun i => cthickening (max 0 (r₂ i)) (s i)) atTop p
tactic: simp_rw [← cthickening_max_zero (r₁ _), ← cthickening_max_zero (r₂ _)]
state_before: α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
⊢ blimsup (fun i => cthickening (max 0 (r₁ i)) (s i)) atTop p ≤ᶠ[ae μ]
    blimsup (fun i => cthickening (max 0 (r₂ i)) (s i)) atTop p
state_after: case inl
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
hM' : 1 ≤ M
⊢ blimsup (fun i => cthickening (max 0 (r₁ i)) (s i)) atTop p ≤ᶠ[ae μ]
    blimsup (fun i => cthickening (max 0 (r₂ i)) (s i)) atTop p

case inr
α : Type u_1
inst✝⁵ : MetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
M : ℝ
hM : 0 < M
r₁ r₂ : ℕ → ℝ
hr : Tendsto r₁ atTop (𝓝[>] 0)
R₁ : ℕ → ℝ := fun i => max 0 (r₁ i)
R₂ : ℕ → ℝ := fun i => max 0 (r₂ i)
hRp : 0 ≤ R₁
hMr : ∀ᶠ (i : ℕ) in atTop, M * R₁ i ≤ R₂ i
hM' : M < 1
⊢ blimsup (fun i => cthickening (max 0 (r₁ i)) (s i)) atTop p ≤ᶠ[ae μ]
    blimsup (fun i => cthickening (max 0 (r₂ i)) (s i)) atTop p
tactic: rcases le_or_lt 1 M with hM' | hM'