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

tactic: borelize β
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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
state_after: α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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: refine aestronglyMeasurable_iff_aemeasurable_separable.2 ⟨?_, ?_⟩
state_before: α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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
state_after: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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 μ

case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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: exact aemeasurable_of_tendsto_metrizable_ae _ (fun n => (hf n).aemeasurable) lim
state_before: case refine_1
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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 μ
state_after: no goals

tactic: rcases u.exists_seq_tendsto with ⟨v, hv⟩
state_before: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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
state_after: case refine_2.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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: have : ∀ n : ℕ, ∃ t : Set β, IsSeparable t ∧ f (v n) ⁻¹' t ∈ ae μ := fun n =>
  (aestronglyMeasurable_iff_aemeasurable_separable.1 (hf (v n))).2
state_before: case refine_2.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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
state_after: case refine_2.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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: choose t t_sep ht using this
state_before: case refine_2.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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
state_after: case refine_2.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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: refine ⟨closure (⋃ i, t i), .closure <| .iUnion t_sep, ?_⟩
state_before: case refine_2.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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
state_after: case refine_2.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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, t i)

tactic: filter_upwards [ae_all_iff.2 ht, lim] with x hx h'x
state_before: case refine_2.intro
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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, t i)
state_after: case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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, t i)

tactic: apply mem_closure_of_tendsto (h'x.comp hv)
state_before: case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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, t i)
state_after: case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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, t i

tactic: filter_upwards with n using mem_iUnion_of_mem n (hx n)
state_before: case h
α : Type u_1
β : Type u_2
γ : Type u_3
ι✝ : Type u_4
inst✝⁵ : Countable ι✝
m : MeasurableSpace α
μ ν : Measure α
inst✝⁴ : TopologicalSpace β
inst✝³ : TopologicalSpace γ
f✝ g✝ : α → β
ι : Type u_5
inst✝² : PseudoMetrizableSpace β
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
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, t i
state_after: no goals