l3lab/ntp-mathlib huggingface dataset:
theorem aemeasurable_of_tendsto_metrizable_ae {ι} {μ : Measure α} {f : ι → α → β} {g : α → β}
    (u : Filter ι) [hu : NeBot u] [IsCountablyGenerated u] (hf : ∀ n, AEMeasurable (f n) μ)
    (h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))) : AEMeasurable g μ := by
  rcases u.exists_seq_tendsto with ⟨v, hv⟩
  have h'f : ∀ n, AEMeasurable (f (v n)) μ := fun n => hf (v n)
  set p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))
  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
  set aeSeqLim := fun x => ite (x ∈ aeSeqSet h'f p) (g x) (⟨f (v 0) x⟩ : Nonempty β).some
  refine'
    ⟨aeSeqLim,
      measurable_of_tendsto_metrizable' atTop (aeSeq.measurable h'f p)
        (tendsto_pi_nhds.mpr fun x => _),
      _⟩
  · simp_rw [aeSeq]
    split_ifs with hx
    · simp_rw [aeSeq.mk_eq_fun_of_mem_aeSeqSet h'f hx]
      

Proof state:
case pos
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 : NeBot u
inst✝ : IsCountablyGenerated u
hf : ∀ (n : ι), AEMeasurable (f n)
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))
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 Nonempty.some (_ : Nonempty β)
x : α
hx : x ∈ aeSeqSet h'f fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))
⊢ Tendsto (fun i => f (v i) x) atTop (𝓝 (g x))

Next tactic:
exact @aeSeq.fun_prop_of_mem_aeSeqSet _ α β _ _ _ _ _ h'f x hx
theorem aemeasurable_of_tendsto_metrizable_ae {ι} {μ : Measure α} {f : ι → α → β} {g : α → β}
    (u : Filter ι) [hu : NeBot u] [IsCountablyGenerated u] (hf : ∀ n, AEMeasurable (f n) μ)
    (h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))) : AEMeasurable g μ := by
  rcases u.exists_seq_tendsto with ⟨v, hv⟩
  have h'f : ∀ n, AEMeasurable (f (v n)) μ := fun n => hf (v n)
  set p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))
  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
  set aeSeqLim := fun x => ite (x ∈ aeSeqSet h'f p) (g x) (⟨f (v 0) x⟩ : Nonempty β).some
  refine'
    ⟨aeSeqLim,
      measurable_of_tendsto_metrizable' atTop (aeSeq.measurable h'f p)
        (tendsto_pi_nhds.mpr fun x => _),
      _⟩
  · simp_rw [aeSeq]
    split_ifs with hx
    · simp_rw [aeSeq.mk_eq_fun_of_mem_aeSeqSet h'f hx]
      exact @aeSeq.fun_prop_of_mem_aeSeqSet _ α β _ _ _ _ _ h'f x hx
    · 

Proof state:
case neg
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 : NeBot u
inst✝ : IsCountablyGenerated u
hf : ∀ (n : ι), AEMeasurable (f n)
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))
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 Nonempty.some (_ : Nonempty β)
x : α
hx : x ∉ aeSeqSet h'f fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))
⊢ Tendsto (fun i => Nonempty.some (_ : Nonempty β)) atTop (𝓝 (Nonempty.some (_ : Nonempty β)))

Next tactic:
exact tendsto_const_nhds
theorem aemeasurable_of_tendsto_metrizable_ae {ι} {μ : Measure α} {f : ι → α → β} {g : α → β}
    (u : Filter ι) [hu : NeBot u] [IsCountablyGenerated u] (hf : ∀ n, AEMeasurable (f n) μ)
    (h_tendsto : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) u (𝓝 (g x))) : AEMeasurable g μ := by
  rcases u.exists_seq_tendsto with ⟨v, hv⟩
  have h'f : ∀ n, AEMeasurable (f (v n)) μ := fun n => hf (v n)
  set p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))
  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
  set aeSeqLim := fun x => ite (x ∈ aeSeqSet h'f p) (g x) (⟨f (v 0) x⟩ : Nonempty β).some
  refine'
    ⟨aeSeqLim,
      measurable_of_tendsto_metrizable' atTop (aeSeq.measurable h'f p)
        (tendsto_pi_nhds.mpr fun x => _),
      _⟩
  · simp_rw [aeSeq]
    split_ifs with hx
    · simp_rw [aeSeq.mk_eq_fun_of_mem_aeSeqSet h'f hx]
      exact @aeSeq.fun_prop_of_mem_aeSeqSet _ α β _ _ _ _ _ h'f x hx
    · exact tendsto_const_nhds
  · 

Proof state:
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 : NeBot u
inst✝ : IsCountablyGenerated u
hf : ∀ (n : ι), AEMeasurable (f n)
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))
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 Nonempty.some (_ : Nonempty β)
⊢ g =ᵐ[μ] aeSeqLim

Next tactic:
exact
      (ite_ae_eq_of_measure_compl_zero g (fun x => (⟨f (v 0) x⟩ : Nonempty β).some) (aeSeqSet h'f p)
          (aeSeq.measure_compl_aeSeqSet_eq_zero h'f hp)).symm
theorem aemeasurable_of_unif_approx {β} [MeasurableSpace β] [PseudoMetricSpace β] [BorelSpace β]
    {μ : Measure α} {g : α → β}
    (hf : ∀ ε > (0 : ℝ), ∃ f : α → β, AEMeasurable f μ ∧ ∀ᵐ x ∂μ, dist (f x) (g x) ≤ ε) :
    AEMeasurable g μ := by
  

Proof state:
α : Type u_1
β✝ : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β✝
inst✝⁵ : PseudoMetrizableSpace β✝
inst✝⁴ : MeasurableSpace β✝
inst✝³ : BorelSpace β✝
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : PseudoMetricSpace β
inst✝ : BorelSpace β
μ : Measure α
g : α → β
hf : ∀ ε > 0, ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε
⊢ AEMeasurable g

Next tactic:
obtain ⟨u, -, u_pos, u_lim⟩ :
    ∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
    exists_seq_strictAnti_tendsto (0 : ℝ)
theorem aemeasurable_of_unif_approx {β} [MeasurableSpace β] [PseudoMetricSpace β] [BorelSpace β]
    {μ : Measure α} {g : α → β}
    (hf : ∀ ε > (0 : ℝ), ∃ f : α → β, AEMeasurable f μ ∧ ∀ᵐ x ∂μ, dist (f x) (g x) ≤ ε) :
    AEMeasurable g μ := by
  obtain ⟨u, -, u_pos, u_lim⟩ :
    ∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
    exists_seq_strictAnti_tendsto (0 : ℝ)
  

Proof state:
case intro.intro.intro
α : Type u_1
β✝ : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β✝
inst✝⁵ : PseudoMetrizableSpace β✝
inst✝⁴ : MeasurableSpace β✝
inst✝³ : BorelSpace β✝
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : PseudoMetricSpace β
inst✝ : BorelSpace β
μ : Measure α
g : α → β
hf : ∀ ε > 0, ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
⊢ AEMeasurable g

Next tactic:
choose f Hf using fun n : ℕ => hf (u n) (u_pos n)
theorem aemeasurable_of_unif_approx {β} [MeasurableSpace β] [PseudoMetricSpace β] [BorelSpace β]
    {μ : Measure α} {g : α → β}
    (hf : ∀ ε > (0 : ℝ), ∃ f : α → β, AEMeasurable f μ ∧ ∀ᵐ x ∂μ, dist (f x) (g x) ≤ ε) :
    AEMeasurable g μ := by
  obtain ⟨u, -, u_pos, u_lim⟩ :
    ∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
    exists_seq_strictAnti_tendsto (0 : ℝ)
  choose f Hf using fun n : ℕ => hf (u n) (u_pos n)
  

Proof state:
case intro.intro.intro
α : Type u_1
β✝ : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β✝
inst✝⁵ : PseudoMetrizableSpace β✝
inst✝⁴ : MeasurableSpace β✝
inst✝³ : BorelSpace β✝
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : PseudoMetricSpace β
inst✝ : BorelSpace β
μ : Measure α
g : α → β
hf : ∀ ε > 0, ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
f : ℕ → α → β
Hf : ∀ (n : ℕ), AEMeasurable (f n) ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n
⊢ AEMeasurable g

Next tactic:
have : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) := by
    have : ∀ᵐ x ∂μ, ∀ n, dist (f n x) (g x) ≤ u n := ae_all_iff.2 fun n => (Hf n).2
    filter_upwards [this]
    intro x hx
    rw [tendsto_iff_dist_tendsto_zero]
    exact squeeze_zero (fun n => dist_nonneg) hx u_lim
theorem aemeasurable_of_unif_approx {β} [MeasurableSpace β] [PseudoMetricSpace β] [BorelSpace β]
    {μ : Measure α} {g : α → β}
    (hf : ∀ ε > (0 : ℝ), ∃ f : α → β, AEMeasurable f μ ∧ ∀ᵐ x ∂μ, dist (f x) (g x) ≤ ε) :
    AEMeasurable g μ := by
  obtain ⟨u, -, u_pos, u_lim⟩ :
    ∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
    exists_seq_strictAnti_tendsto (0 : ℝ)
  choose f Hf using fun n : ℕ => hf (u n) (u_pos n)
  have : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) := by
    

Proof state:
α : Type u_1
β✝ : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β✝
inst✝⁵ : PseudoMetrizableSpace β✝
inst✝⁴ : MeasurableSpace β✝
inst✝³ : BorelSpace β✝
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : PseudoMetricSpace β
inst✝ : BorelSpace β
μ : Measure α
g : α → β
hf : ∀ ε > 0, ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
f : ℕ → α → β
Hf : ∀ (n : ℕ), AEMeasurable (f n) ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))

Next tactic:
have : ∀ᵐ x ∂μ, ∀ n, dist (f n x) (g x) ≤ u n := ae_all_iff.2 fun n => (Hf n).2
theorem aemeasurable_of_unif_approx {β} [MeasurableSpace β] [PseudoMetricSpace β] [BorelSpace β]
    {μ : Measure α} {g : α → β}
    (hf : ∀ ε > (0 : ℝ), ∃ f : α → β, AEMeasurable f μ ∧ ∀ᵐ x ∂μ, dist (f x) (g x) ≤ ε) :
    AEMeasurable g μ := by
  obtain ⟨u, -, u_pos, u_lim⟩ :
    ∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
    exists_seq_strictAnti_tendsto (0 : ℝ)
  choose f Hf using fun n : ℕ => hf (u n) (u_pos n)
  have : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) := by
    have : ∀ᵐ x ∂μ, ∀ n, dist (f n x) (g x) ≤ u n := ae_all_iff.2 fun n => (Hf n).2
    

Proof state:
α : Type u_1
β✝ : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β✝
inst✝⁵ : PseudoMetrizableSpace β✝
inst✝⁴ : MeasurableSpace β✝
inst✝³ : BorelSpace β✝
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : PseudoMetricSpace β
inst✝ : BorelSpace β
μ : Measure α
g : α → β
hf : ∀ ε > 0, ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
f : ℕ → α → β
Hf : ∀ (n : ℕ), AEMeasurable (f n) ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n
this : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), dist (f n x) (g x) ≤ u n
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))

Next tactic:
filter_upwards [this]
theorem aemeasurable_of_unif_approx {β} [MeasurableSpace β] [PseudoMetricSpace β] [BorelSpace β]
    {μ : Measure α} {g : α → β}
    (hf : ∀ ε > (0 : ℝ), ∃ f : α → β, AEMeasurable f μ ∧ ∀ᵐ x ∂μ, dist (f x) (g x) ≤ ε) :
    AEMeasurable g μ := by
  obtain ⟨u, -, u_pos, u_lim⟩ :
    ∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
    exists_seq_strictAnti_tendsto (0 : ℝ)
  choose f Hf using fun n : ℕ => hf (u n) (u_pos n)
  have : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) := by
    have : ∀ᵐ x ∂μ, ∀ n, dist (f n x) (g x) ≤ u n := ae_all_iff.2 fun n => (Hf n).2
    filter_upwards [this]
    

Proof state:
case h
α : Type u_1
β✝ : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β✝
inst✝⁵ : PseudoMetrizableSpace β✝
inst✝⁴ : MeasurableSpace β✝
inst✝³ : BorelSpace β✝
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : PseudoMetricSpace β
inst✝ : BorelSpace β
μ : Measure α
g : α → β
hf : ∀ ε > 0, ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
f : ℕ → α → β
Hf : ∀ (n : ℕ), AEMeasurable (f n) ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n
this : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), dist (f n x) (g x) ≤ u n
⊢ ∀ (a : α), (∀ (n : ℕ), dist (f n a) (g a) ≤ u n) → Tendsto (fun n => f n a) atTop (𝓝 (g a))

Next tactic:
intro x hx
theorem aemeasurable_of_unif_approx {β} [MeasurableSpace β] [PseudoMetricSpace β] [BorelSpace β]
    {μ : Measure α} {g : α → β}
    (hf : ∀ ε > (0 : ℝ), ∃ f : α → β, AEMeasurable f μ ∧ ∀ᵐ x ∂μ, dist (f x) (g x) ≤ ε) :
    AEMeasurable g μ := by
  obtain ⟨u, -, u_pos, u_lim⟩ :
    ∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
    exists_seq_strictAnti_tendsto (0 : ℝ)
  choose f Hf using fun n : ℕ => hf (u n) (u_pos n)
  have : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) := by
    have : ∀ᵐ x ∂μ, ∀ n, dist (f n x) (g x) ≤ u n := ae_all_iff.2 fun n => (Hf n).2
    filter_upwards [this]
    intro x hx
    

Proof state:
case h
α : Type u_1
β✝ : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β✝
inst✝⁵ : PseudoMetrizableSpace β✝
inst✝⁴ : MeasurableSpace β✝
inst✝³ : BorelSpace β✝
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : PseudoMetricSpace β
inst✝ : BorelSpace β
μ : Measure α
g : α → β
hf : ∀ ε > 0, ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
f : ℕ → α → β
Hf : ∀ (n : ℕ), AEMeasurable (f n) ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n
this : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), dist (f n x) (g x) ≤ u n
x : α
hx : ∀ (n : ℕ), dist (f n x) (g x) ≤ u n
⊢ Tendsto (fun n => f n x) atTop (𝓝 (g x))

Next tactic:
rw [tendsto_iff_dist_tendsto_zero]
theorem aemeasurable_of_unif_approx {β} [MeasurableSpace β] [PseudoMetricSpace β] [BorelSpace β]
    {μ : Measure α} {g : α → β}
    (hf : ∀ ε > (0 : ℝ), ∃ f : α → β, AEMeasurable f μ ∧ ∀ᵐ x ∂μ, dist (f x) (g x) ≤ ε) :
    AEMeasurable g μ := by
  obtain ⟨u, -, u_pos, u_lim⟩ :
    ∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
    exists_seq_strictAnti_tendsto (0 : ℝ)
  choose f Hf using fun n : ℕ => hf (u n) (u_pos n)
  have : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) := by
    have : ∀ᵐ x ∂μ, ∀ n, dist (f n x) (g x) ≤ u n := ae_all_iff.2 fun n => (Hf n).2
    filter_upwards [this]
    intro x hx
    rw [tendsto_iff_dist_tendsto_zero]
    

Proof state:
case h
α : Type u_1
β✝ : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β✝
inst✝⁵ : PseudoMetrizableSpace β✝
inst✝⁴ : MeasurableSpace β✝
inst✝³ : BorelSpace β✝
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : PseudoMetricSpace β
inst✝ : BorelSpace β
μ : Measure α
g : α → β
hf : ∀ ε > 0, ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
f : ℕ → α → β
Hf : ∀ (n : ℕ), AEMeasurable (f n) ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n
this : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), dist (f n x) (g x) ≤ u n
x : α
hx : ∀ (n : ℕ), dist (f n x) (g x) ≤ u n
⊢ Tendsto (fun b => dist (f b x) (g x)) atTop (𝓝 0)

Next tactic:
exact squeeze_zero (fun n => dist_nonneg) hx u_lim
theorem aemeasurable_of_unif_approx {β} [MeasurableSpace β] [PseudoMetricSpace β] [BorelSpace β]
    {μ : Measure α} {g : α → β}
    (hf : ∀ ε > (0 : ℝ), ∃ f : α → β, AEMeasurable f μ ∧ ∀ᵐ x ∂μ, dist (f x) (g x) ≤ ε) :
    AEMeasurable g μ := by
  obtain ⟨u, -, u_pos, u_lim⟩ :
    ∃ u : ℕ → ℝ, StrictAnti u ∧ (∀ n : ℕ, 0 < u n) ∧ Tendsto u atTop (𝓝 0) :=
    exists_seq_strictAnti_tendsto (0 : ℝ)
  choose f Hf using fun n : ℕ => hf (u n) (u_pos n)
  have : ∀ᵐ x ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x)) := by
    have : ∀ᵐ x ∂μ, ∀ n, dist (f n x) (g x) ≤ u n := ae_all_iff.2 fun n => (Hf n).2
    filter_upwards [this]
    intro x hx
    rw [tendsto_iff_dist_tendsto_zero]
    exact squeeze_zero (fun n => dist_nonneg) hx u_lim
  

Proof state:
case intro.intro.intro
α : Type u_1
β✝ : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β✝
inst✝⁵ : PseudoMetrizableSpace β✝
inst✝⁴ : MeasurableSpace β✝
inst✝³ : BorelSpace β✝
β : Type u_3
inst✝² : MeasurableSpace β
inst✝¹ : PseudoMetricSpace β
inst✝ : BorelSpace β
μ : Measure α
g : α → β
hf : ∀ ε > 0, ∃ f, AEMeasurable f ∧ ∀ᵐ (x : α) ∂μ, dist (f x) (g x) ≤ ε
u : ℕ → ℝ
u_pos : ∀ (n : ℕ), 0 < u n
u_lim : Tendsto u atTop (𝓝 0)
f : ℕ → α → β
Hf : ∀ (n : ℕ), AEMeasurable (f n) ∧ ∀ᵐ (x : α) ∂μ, dist (f n x) (g x) ≤ u n
this : ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (g x))
⊢ AEMeasurable g

Next tactic:
exact aemeasurable_of_tendsto_metrizable_ae' (fun n => (Hf n).1) this
theorem measurable_limit_of_tendsto_metrizable_ae {ι} [Countable ι] [Nonempty ι] {μ : Measure α}
    {f : ι → α → β} {L : Filter ι} [L.IsCountablyGenerated] (hf : ∀ n, AEMeasurable (f n) μ)
    (h_ae_tendsto : ∀ᵐ x ∂μ, ∃ l : β, Tendsto (fun n => f n x) L (𝓝 l)) :
    ∃ f_lim : α → β, Measurable f_lim ∧ ∀ᵐ x ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) := by
  

Proof state:
α : Type u_1
β : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_3
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
⊢ ∃ f_lim, Measurable f_lim ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))

Next tactic:
inhabit ι
theorem measurable_limit_of_tendsto_metrizable_ae {ι} [Countable ι] [Nonempty ι] {μ : Measure α}
    {f : ι → α → β} {L : Filter ι} [L.IsCountablyGenerated] (hf : ∀ n, AEMeasurable (f n) μ)
    (h_ae_tendsto : ∀ᵐ x ∂μ, ∃ l : β, Tendsto (fun n => f n x) L (𝓝 l)) :
    ∃ f_lim : α → β, Measurable f_lim ∧ ∀ᵐ x ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) := by
  inhabit ι
  

Proof state:
α : Type u_1
β : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_3
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
⊢ ∃ f_lim, Measurable f_lim ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))

Next tactic:
rcases eq_or_neBot L with (rfl | hL)
theorem measurable_limit_of_tendsto_metrizable_ae {ι} [Countable ι] [Nonempty ι] {μ : Measure α}
    {f : ι → α → β} {L : Filter ι} [L.IsCountablyGenerated] (hf : ∀ n, AEMeasurable (f n) μ)
    (h_ae_tendsto : ∀ᵐ x ∂μ, ∃ l : β, Tendsto (fun n => f n x) L (𝓝 l)) :
    ∃ f_lim : α → β, Measurable f_lim ∧ ∀ᵐ x ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) := by
  inhabit ι
  rcases eq_or_neBot L with (rfl | hL)
  · 

Proof state:
case inl
α : Type u_1
β : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_3
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : Measure α
f : ι → α → β
hf : ∀ (n : ι), AEMeasurable (f n)
inhabited_h : Inhabited ι
inst✝ : IsCountablyGenerated ⊥
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) ⊥ (𝓝 l)
⊢ ∃ f_lim, Measurable f_lim ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) ⊥ (𝓝 (f_lim x))

Next tactic:
exact ⟨(hf default).mk _, (hf default).measurable_mk, eventually_of_forall fun x => tendsto_bot⟩
theorem measurable_limit_of_tendsto_metrizable_ae {ι} [Countable ι] [Nonempty ι] {μ : Measure α}
    {f : ι → α → β} {L : Filter ι} [L.IsCountablyGenerated] (hf : ∀ n, AEMeasurable (f n) μ)
    (h_ae_tendsto : ∀ᵐ x ∂μ, ∃ l : β, Tendsto (fun n => f n x) L (𝓝 l)) :
    ∃ f_lim : α → β, Measurable f_lim ∧ ∀ᵐ x ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) := by
  inhabit ι
  rcases eq_or_neBot L with (rfl | hL)
  · exact ⟨(hf default).mk _, (hf default).measurable_mk, eventually_of_forall fun x => tendsto_bot⟩
  

Proof state:
case inr
α : Type u_1
β : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_3
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : NeBot L
⊢ ∃ f_lim, Measurable f_lim ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))

Next tactic:
let p : α → (ι → β) → Prop := fun x f' => ∃ l : β, Tendsto (fun n => f' n) L (𝓝 l)
theorem measurable_limit_of_tendsto_metrizable_ae {ι} [Countable ι] [Nonempty ι] {μ : Measure α}
    {f : ι → α → β} {L : Filter ι} [L.IsCountablyGenerated] (hf : ∀ n, AEMeasurable (f n) μ)
    (h_ae_tendsto : ∀ᵐ x ∂μ, ∃ l : β, Tendsto (fun n => f n x) L (𝓝 l)) :
    ∃ f_lim : α → β, Measurable f_lim ∧ ∀ᵐ x ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) := by
  inhabit ι
  rcases eq_or_neBot L with (rfl | hL)
  · exact ⟨(hf default).mk _, (hf default).measurable_mk, eventually_of_forall fun x => tendsto_bot⟩
  let p : α → (ι → β) → Prop := fun x f' => ∃ l : β, Tendsto (fun n => f' n) L (𝓝 l)
  

Proof state:
case inr
α : Type u_1
β : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_3
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : NeBot L
p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)
⊢ ∃ f_lim, Measurable f_lim ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))

Next tactic:
have hp_mem : ∀ x ∈ aeSeqSet hf p, p x fun n => f n x := fun x hx =>
    aeSeq.fun_prop_of_mem_aeSeqSet hf hx
theorem measurable_limit_of_tendsto_metrizable_ae {ι} [Countable ι] [Nonempty ι] {μ : Measure α}
    {f : ι → α → β} {L : Filter ι} [L.IsCountablyGenerated] (hf : ∀ n, AEMeasurable (f n) μ)
    (h_ae_tendsto : ∀ᵐ x ∂μ, ∃ l : β, Tendsto (fun n => f n x) L (𝓝 l)) :
    ∃ f_lim : α → β, Measurable f_lim ∧ ∀ᵐ x ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x)) := by
  inhabit ι
  rcases eq_or_neBot L with (rfl | hL)
  · exact ⟨(hf default).mk _, (hf default).measurable_mk, eventually_of_forall fun x => tendsto_bot⟩
  let p : α → (ι → β) → Prop := fun x f' => ∃ l : β, Tendsto (fun n => f' n) L (𝓝 l)
  have hp_mem : ∀ x ∈ aeSeqSet hf p, p x fun n => f n x := fun x hx =>
    aeSeq.fun_prop_of_mem_aeSeqSet hf hx
  

Proof state:
case inr
α : Type u_1
β : Type u_2
inst✝⁷ : MeasurableSpace α
inst✝⁶ : TopologicalSpace β
inst✝⁵ : PseudoMetrizableSpace β
inst✝⁴ : MeasurableSpace β
inst✝³ : BorelSpace β
ι : Type u_3
inst✝² : Countable ι
inst✝¹ : Nonempty ι
μ : Measure α
f : ι → α → β
L : Filter ι
inst✝ : IsCountablyGenerated L
hf : ∀ (n : ι), AEMeasurable (f n)
h_ae_tendsto : ∀ᵐ (x : α) ∂μ, ∃ l, Tendsto (fun n => f n x) L (𝓝 l)
inhabited_h : Inhabited ι
hL : NeBot L
p : α → (ι → β) → Prop := fun x f' => ∃ l, Tendsto (fun n => f' n) L (𝓝 l)
hp_mem : ∀ x ∈ aeSeqSet hf p, p x fun n => f n x
⊢ ∃ f_lim, Measurable f_lim ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun n => f n x) L (𝓝 (f_lim x))

Next tactic:
have h_ae_eq : ∀ᵐ x ∂μ, ∀ n, aeSeq hf p n x = f n x := aeSeq.aeSeq_eq_fun_ae hf h_ae_tendsto