Theorem: MeasureTheory.Egorov.measure_notConvergentSeq_tendsto_zero
File: Mathlib/MeasureTheory/Function/Egorov.lean

tactic: cases' isEmpty_or_nonempty ι with h h
state_before: α : Type u_2
β : Type u_3
ι : Type u_1
m : MeasurableSpace α
inst✝² : MetricSpace β
μ : Measure α
n✝ : ℕ
i j : ι
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : ↑↑μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
n : ℕ
⊢ Tendsto (fun j => ↑↑μ (s ∩ notConvergentSeq f g n j)) atTop (𝓝 0)

tactic: rw [← measure_inter_notConvergentSeq_eq_zero hfg n, Set.inter_iInter]
state_before: case inr
α : Type u_2
β : Type u_3
ι : Type u_1
m : MeasurableSpace α
inst✝² : MetricSpace β
μ : Measure α
n✝ : ℕ
i j : ι
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : ↑↑μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
n : ℕ
h : Nonempty ι
⊢ Tendsto (fun j => ↑↑μ (s ∩ notConvergentSeq f g n j)) atTop (𝓝 0)

tactic: refine' tendsto_measure_iInter (fun n => hsm.inter <| notConvergentSeq_measurableSet hf hg)
  (fun k l hkl => Set.inter_subset_inter_right _ <| notConvergentSeq_antitone hkl)
  ⟨h.some,
    (lt_of_le_of_lt (measure_mono <| Set.inter_subset_left _ _) (lt_top_iff_ne_top.2 hs)).ne⟩
state_before: case inr
α : Type u_2
β : Type u_3
ι : Type u_1
m : MeasurableSpace α
inst✝² : MetricSpace β
μ : Measure α
n✝ : ℕ
i j : ι
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : ↑↑μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
n : ℕ
h : Nonempty ι
⊢ Tendsto (fun j => ↑↑μ (s ∩ notConvergentSeq f g n j)) atTop
    (𝓝 (↑↑μ (⋂ (i : ι), s ∩ notConvergentSeq (fun n x => f n x) (fun x => g x) n i)))

tactic: have : (fun j => μ (s ∩ notConvergentSeq f g n j)) = fun j => 0 := by
  simp only [eq_iff_true_of_subsingleton]
state_before: case inl
α : Type u_2
β : Type u_3
ι : Type u_1
m : MeasurableSpace α
inst✝² : MetricSpace β
μ : Measure α
n✝ : ℕ
i j : ι
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : ↑↑μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
n : ℕ
h : IsEmpty ι
⊢ Tendsto (fun j => ↑↑μ (s ∩ notConvergentSeq f g n j)) atTop (𝓝 0)

tactic: rw [this]
state_before: case inl
α : Type u_2
β : Type u_3
ι : Type u_1
m : MeasurableSpace α
inst✝² : MetricSpace β
μ : Measure α
n✝ : ℕ
i j : ι
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : ↑↑μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
n : ℕ
h : IsEmpty ι
this : (fun j => ↑↑μ (s ∩ notConvergentSeq f g n j)) = fun j => 0
⊢ Tendsto (fun j => ↑↑μ (s ∩ notConvergentSeq f g n j)) atTop (𝓝 0)

tactic: exact tendsto_const_nhds
state_before: case inl
α : Type u_2
β : Type u_3
ι : Type u_1
m : MeasurableSpace α
inst✝² : MetricSpace β
μ : Measure α
n✝ : ℕ
i j : ι
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : ↑↑μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
n : ℕ
h : IsEmpty ι
this : (fun j => ↑↑μ (s ∩ notConvergentSeq f g n j)) = fun j => 0
⊢ Tendsto (fun j => 0) atTop (𝓝 0)

tactic: simp only [eq_iff_true_of_subsingleton]
state_before: α : Type u_2
β : Type u_3
ι : Type u_1
m : MeasurableSpace α
inst✝² : MetricSpace β
μ : Measure α
n✝ : ℕ
i j : ι
s : Set α
ε : ℝ
f : ι → α → β
g : α → β
inst✝¹ : SemilatticeSup ι
inst✝ : Countable ι
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hsm : MeasurableSet s
hs : ↑↑μ s ≠ ⊤
hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))
n : ℕ
h : IsEmpty ι
⊢ (fun j => ↑↑μ (s ∩ notConvergentSeq f g n j)) = fun j => 0