Theorem: aemeasurable_of_tendsto_metrizable_ae
File: Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean

Proof steps:
state_before: α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : PseudoMetrizableSpace β
inst✝² : MeasurableSpace β
inst✝¹ : BorelSpace β
ι : Type u_3
μ : Measure α
f : ι → α → β
g : α → β
u : Filter ι
hu : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
⊢ AEMeasurable g μ
state_after: case intro
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : PseudoMetrizableSpace β
inst✝² : MeasurableSpace β
inst✝¹ : BorelSpace β
ι : Type u_3
μ : Measure α
f : ι → α → β
g : α → β
u : Filter ι
hu : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
v : ℕ → ι
hv : Tendsto v atTop u
⊢ AEMeasurable g μ
tactic: rcases u.exists_seq_tendsto with ⟨v, hv⟩
state_before: case intro
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : PseudoMetrizableSpace β
inst✝² : MeasurableSpace β
inst✝¹ : BorelSpace β
ι : Type u_3
μ : Measure α
f : ι → α → β
g : α → β
u : Filter ι
hu : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
v : ℕ → ι
hv : Tendsto v atTop u
⊢ AEMeasurable g μ
state_after: case intro
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : PseudoMetrizableSpace β
inst✝² : MeasurableSpace β
inst✝¹ : BorelSpace β
ι : Type u_3
μ : Measure α
f : ι → α → β
g : α → β
u : Filter ι
hu : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
v : ℕ → ι
hv : Tendsto v atTop u
h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) μ
⊢ AEMeasurable g μ
tactic: have h'f : ∀ n, AEMeasurable (f (v n)) μ := fun n => hf (v n)
state_before: case intro
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : PseudoMetrizableSpace β
inst✝² : MeasurableSpace β
inst✝¹ : BorelSpace β
ι : Type u_3
μ : Measure α
f : ι → α → β
g : α → β
u : Filter ι
hu : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
v : ℕ → ι
hv : Tendsto v atTop u
h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) μ
⊢ AEMeasurable g μ
state_after: case intro
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : PseudoMetrizableSpace β
inst✝² : MeasurableSpace β
inst✝¹ : BorelSpace β
ι : Type u_3
μ : Measure α
f : ι → α → β
g : α → β
u : Filter ι
hu : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
v : ℕ → ι
hv : Tendsto v atTop u
h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) μ
p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))
⊢ AEMeasurable g μ
tactic: set p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))
state_before: case intro
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : PseudoMetrizableSpace β
inst✝² : MeasurableSpace β
inst✝¹ : BorelSpace β
ι : Type u_3
μ : Measure α
f : ι → α → β
g : α → β
u : Filter ι
hu : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
v : ℕ → ι
hv : Tendsto v atTop u
h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) μ
p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))
⊢ AEMeasurable g μ
state_after: case intro
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : PseudoMetrizableSpace β
inst✝² : MeasurableSpace β
inst✝¹ : BorelSpace β
ι : Type u_3
μ : Measure α
f : ι → α → β
g : α → β
u : Filter ι
hu : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
v : ℕ → ι
hv : Tendsto v atTop u
h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) μ
p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))
hp : ∀ᵐ (x : α) ∂μ, p x fun n => f (v n) x
⊢ AEMeasurable g μ
tactic: have hp : ∀ᵐ x ∂μ, p x fun n => f (v n) x := by
    filter_upwards [h_tendsto] with x hx using hx.comp hv
state_before: case intro
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : PseudoMetrizableSpace β
inst✝² : MeasurableSpace β
inst✝¹ : BorelSpace β
ι : Type u_3
μ : Measure α
f : ι → α → β
g : α → β
u : Filter ι
hu : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
v : ℕ → ι
hv : Tendsto v atTop u
h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) μ
p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))
hp : ∀ᵐ (x : α) ∂μ, p x fun n => f (v n) x
⊢ AEMeasurable g μ
state_after: case intro
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : PseudoMetrizableSpace β
inst✝² : MeasurableSpace β
inst✝¹ : BorelSpace β
ι : Type u_3
μ : Measure α
f : ι → α → β
g : α → β
u : Filter ι
hu : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
v : ℕ → ι
hv : Tendsto v atTop u
h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) μ
p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))
hp : ∀ᵐ (x : α) ∂μ, p x fun n => f (v n) x
aeSeqLim : α → β := fun x => if x ∈ aeSeqSet h'f p then g x else ⋯.some
⊢ AEMeasurable g μ
tactic: set aeSeqLim := fun x => ite (x ∈ aeSeqSet h'f p) (g x) (⟨f (v 0) x⟩ : Nonempty β).some
state_before: case intro
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : PseudoMetrizableSpace β
inst✝² : MeasurableSpace β
inst✝¹ : BorelSpace β
ι : Type u_3
μ : Measure α
f : ι → α → β
g : α → β
u : Filter ι
hu : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
v : ℕ → ι
hv : Tendsto v atTop u
h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) μ
p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))
hp : ∀ᵐ (x : α) ∂μ, p x fun n => f (v n) x
aeSeqLim : α → β := fun x => if x ∈ aeSeqSet h'f p then g x else ⋯.some
⊢ AEMeasurable g μ
state_after: case intro.refine_1
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : PseudoMetrizableSpace β
inst✝² : MeasurableSpace β
inst✝¹ : BorelSpace β
ι : Type u_3
μ : Measure α
f : ι → α → β
g : α → β
u : Filter ι
hu : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
v : ℕ → ι
hv : Tendsto v atTop u
h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) μ
p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))
hp : ∀ᵐ (x : α) ∂μ, p x fun n => f (v n) x
aeSeqLim : α → β := fun x => if x ∈ aeSeqSet h'f p then g x else ⋯.some
x : α
⊢ Tendsto (fun i => aeSeq h'f p i x) atTop (𝓝 (aeSeqLim x))

case intro.refine_2
α : Type u_1
β : Type u_2
inst✝⁵ : MeasurableSpace α
inst✝⁴ : TopologicalSpace β
inst✝³ : PseudoMetrizableSpace β
inst✝² : MeasurableSpace β
inst✝¹ : BorelSpace β
ι : Type u_3
μ : Measure α
f : ι → α → β
g : α → β
u : Filter ι
hu : u.NeBot
inst✝ : u.IsCountablyGenerated
hf : ∀ (n : ι), AEMeasurable (f n) μ
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))
v : ℕ → ι
hv : Tendsto v atTop u
h'f : ∀ (n : ℕ), AEMeasurable (f (v n)) μ
p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))
hp : ∀ᵐ (x : α) ∂μ, p x fun n => f (v n) x
aeSeqLim : α → β := fun x => if x ∈ aeSeqSet h'f p then g x else ⋯.some
⊢ g =ᶠ[ae μ] aeSeqLim
tactic: refine
    ⟨aeSeqLim,
      measurable_of_tendsto_metrizable' atTop (aeSeq.measurable h'f p)
        (tendsto_pi_nhds.mpr fun x => ?_),
      ?_⟩