Theorem: aestronglyMeasurable_of_tendsto_ae
File: Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean

tactic: borelize β
state_before: α : Type u_3
β : Type u_2
γ : Type ?u.394279
ι✝ : Type ?u.394282
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_1
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
f : ι → α → β
g : α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
⊢ AEStronglyMeasurable g μ

tactic: refine' aestronglyMeasurable_iff_aemeasurable_separable.2 ⟨_, _⟩
state_before: α : Type u_3
β : Type u_2
γ : Type ?u.394279
ι✝ : Type ?u.394282
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_1
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
f : ι → α → β
g : α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ AEStronglyMeasurable g μ

tactic: exact aemeasurable_of_tendsto_metrizable_ae _ (fun n => (hf n).aemeasurable) lim
state_before: case refine'_1
α : Type u_3
β : Type u_2
γ : Type ?u.394279
ι✝ : Type ?u.394282
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_1
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
f : ι → α → β
g : α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ AEMeasurable g

tactic: rcases u.exists_seq_tendsto with ⟨v, hv⟩
state_before: case refine'_2
α : Type u_3
β : Type u_2
γ : Type ?u.394279
ι✝ : Type ?u.394282
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_1
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
f : ι → α → β
g : α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂μ, g x ∈ t

tactic: have : ∀ n : ℕ, ∃ t : Set β, IsSeparable t ∧ f (v n) ⁻¹' t ∈ μ.ae := fun n =>
  (aestronglyMeasurable_iff_aemeasurable_separable.1 (hf (v n))).2
state_before: case refine'_2.intro
α : Type u_3
β : Type u_2
γ : Type ?u.394279
ι✝ : Type ?u.394282
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_1
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
f : ι → α → β
g : α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
v : ℕ → ι
hv : Tendsto v atTop u
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂μ, g x ∈ t

tactic: choose t t_sep ht using this
state_before: case refine'_2.intro
α : Type u_3
β : Type u_2
γ : Type ?u.394279
ι✝ : Type ?u.394282
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_1
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
f : ι → α → β
g : α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
v : ℕ → ι
hv : Tendsto v atTop u
this : ∀ (n : ℕ), ∃ t, IsSeparable t ∧ f (v n) ⁻¹' t ∈ ae μ
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂μ, g x ∈ t

tactic: refine' ⟨closure (⋃ i, t i), (isSeparable_iUnion fun i => t_sep i).closure, _⟩
state_before: case refine'_2.intro
α : Type u_3
β : Type u_2
γ : Type ?u.394279
ι✝ : Type ?u.394282
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_1
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
f : ι → α → β
g : α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
v : ℕ → ι
hv : Tendsto v atTop u
t : ℕ → Set β
t_sep : ∀ (n : ℕ), IsSeparable (t n)
ht : ∀ (n : ℕ), f (v n) ⁻¹' t n ∈ ae μ
⊢ ∃ t, IsSeparable t ∧ ∀ᵐ (x : α) ∂μ, g x ∈ t

tactic: filter_upwards [ae_all_iff.2 ht, lim] with x hx h'x
state_before: case refine'_2.intro
α : Type u_3
β : Type u_2
γ : Type ?u.394279
ι✝ : Type ?u.394282
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_1
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
f : ι → α → β
g : α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
v : ℕ → ι
hv : Tendsto v atTop u
t : ℕ → Set β
t_sep : ∀ (n : ℕ), IsSeparable (t n)
ht : ∀ (n : ℕ), f (v n) ⁻¹' t n ∈ ae μ
⊢ ∀ᵐ (x : α) ∂μ, g x ∈ closure (⋃ (i : ℕ), t i)

tactic: apply mem_closure_of_tendsto (h'x.comp hv)
state_before: case h
α : Type u_3
β : Type u_2
γ : Type ?u.394279
ι✝ : Type ?u.394282
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_1
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
f : ι → α → β
g : α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
v : ℕ → ι
hv : Tendsto v atTop u
t : ℕ → Set β
t_sep : ∀ (n : ℕ), IsSeparable (t n)
ht : ∀ (n : ℕ), f (v n) ⁻¹' t n ∈ ae μ
x : α
hx : ∀ (i : ℕ), f (v i) x ∈ t i
h'x : Tendsto (fun n => f n x) u (𝓝 (g x))
⊢ g x ∈ closure (⋃ (i : ℕ), t i)

tactic: refine eventually_of_forall fun n => ?_
state_before: case h
α : Type u_3
β : Type u_2
γ : Type ?u.394279
ι✝ : Type ?u.394282
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_1
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
f : ι → α → β
g : α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
v : ℕ → ι
hv : Tendsto v atTop u
t : ℕ → Set β
t_sep : ∀ (n : ℕ), IsSeparable (t n)
ht : ∀ (n : ℕ), f (v n) ⁻¹' t n ∈ ae μ
x : α
hx : ∀ (i : ℕ), f (v i) x ∈ t i
h'x : Tendsto (fun n => f n x) u (𝓝 (g x))
⊢ ∀ᶠ (x_1 : ℕ) in atTop, ((fun n => f n x) ∘ v) x_1 ∈ ⋃ (i : ℕ), t i

tactic: apply mem_iUnion_of_mem n
state_before: case h
α : Type u_3
β : Type u_2
γ : Type ?u.394279
ι✝ : Type ?u.394282
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_1
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
f : ι → α → β
g : α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
v : ℕ → ι
hv : Tendsto v atTop u
t : ℕ → Set β
t_sep : ∀ (n : ℕ), IsSeparable (t n)
ht : ∀ (n : ℕ), f (v n) ⁻¹' t n ∈ ae μ
x : α
hx : ∀ (i : ℕ), f (v i) x ∈ t i
h'x : Tendsto (fun n => f n x) u (𝓝 (g x))
n : ℕ
⊢ ((fun n => f n x) ∘ v) n ∈ ⋃ (i : ℕ), t i

tactic: exact hx n
state_before: case h
α : Type u_3
β : Type u_2
γ : Type ?u.394279
ι✝ : Type ?u.394282
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_1
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : NeBot u
inst✝ : IsCountablyGenerated u
f : ι → α → β
g : α → β
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
v : ℕ → ι
hv : Tendsto v atTop u
t : ℕ → Set β
t_sep : ∀ (n : ℕ), IsSeparable (t n)
ht : ∀ (n : ℕ), f (v n) ⁻¹' t n ∈ ae μ
x : α
hx : ∀ (i : ℕ), f (v i) x ∈ t i
h'x : Tendsto (fun n => f n x) u (𝓝 (g x))
n : ℕ
⊢ ((fun n => f n x) ∘ v) n ∈ t n