-- Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
theorem measurable_uncurry_of_continuous_of_measurable {α β ι : Type*} [TopologicalSpace ι]
    [MetrizableSpace ι] [MeasurableSpace ι] [SecondCountableTopology ι] [OpensMeasurableSpace ι]
    {mβ : MeasurableSpace β} [TopologicalSpace β] [PseudoMetrizableSpace β] [BorelSpace β]
    {m : MeasurableSpace α} {u : ι → α → β} (hu_cont : ∀ x, Continuous fun i => u i x)
    (h : ∀ i, Measurable (u i)) : Measurable (Function.uncurry u)

Proof state:
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
mβ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : BorelSpace β
m : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), Measurable (u i)
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u ((t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p => u ((t_sf n) p.1) p.2
⊢ ∀ (x : ι × α), Tendsto (fun i => U i x) atTop (𝓝 (u x.1 x.2))

Tactic:
exact fun p => ht_sf p.fst p.snd

Result:
no goals

Proof state:
case intro
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
mβ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : BorelSpace β
m : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), Measurable (u i)
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u ((t_sf n) j) x) atTop (𝓝 (u j x))
⊢ Measurable (uncurry u)

Tactic:
let U (n : ℕ) (p : ι × α) := u (t_sf n p.fst) p.snd

Result:
case intro
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
mβ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : BorelSpace β
m : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), Measurable (u i)
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u ((t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p => u ((t_sf n) p.1) p.2
⊢ Measurable (uncurry u)

Proof state:
case intro
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
mβ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : BorelSpace β
m : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), Measurable (u i)
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u ((t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p => u ((t_sf n) p.1) p.2
h_tendsto : Tendsto U atTop (𝓝 fun p => u p.1 p.2)
n : ℕ
h_meas : Measurable fun p => u (↑p.1) p.2
⊢ Measurable (U n)

Tactic:
have :
  (fun p : ι × α => u (t_sf n p.fst) p.snd) =
    (fun p : ↥(t_sf n).range × α => u p.fst p.snd) ∘ fun p : ι × α =>
      (⟨t_sf n p.fst, SimpleFunc.mem_range_self _ _⟩, p.snd) :=
  rfl

Result:
case intro
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
mβ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : BorelSpace β
m : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), Measurable (u i)
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u ((t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p => u ((t_sf n) p.1) p.2
h_tendsto : Tendsto U atTop (𝓝 fun p => u p.1 p.2)
n : ℕ
h_meas : Measurable fun p => u (↑p.1) p.2
this : (fun p => u ((t_sf n) p.1) p.2) = (fun p => u (↑p.1) p.2) ∘ fun p => (⟨(t_sf n) p.1, ⋯⟩, p.2)
⊢ Measurable (U n)

Proof state:
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
mβ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : BorelSpace β
m : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), Measurable (u i)
⊢ ∃ t, ∀ (j : ι) (x : α), Tendsto (fun n => u ((t n) j) x) atTop (𝓝 (u j x))

Tactic:
have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id

Result:
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
mβ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : BorelSpace β
m : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), Measurable (u i)
h_str_meas : StronglyMeasurable id
⊢ ∃ t, ∀ (j : ι) (x : α), Tendsto (fun n => u ((t n) j) x) atTop (𝓝 (u j x))

Proof state:
case intro
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
mβ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : BorelSpace β
m : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), Measurable (u i)
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u ((t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p => u ((t_sf n) p.1) p.2
⊢ Measurable (uncurry u)

Tactic:
have h_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd) := by
  rw [tendsto_pi_nhds]
  exact fun p => ht_sf p.fst p.snd

Result:
case intro
α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
mβ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : BorelSpace β
m : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous fun i => u i x
h : ∀ (i : ι), Measurable (u i)
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u ((t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p => u ((t_sf n) p.1) p.2
h_tendsto : Tendsto U atTop (𝓝 fun p => u p.1 p.2)
⊢ Measurable (uncurry u)