/-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
  Fubini's theorem is measurable.
  This version has `f` in curried form. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right [SigmaFinite ν] ⦃f : α → β → E⦄
    (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂ν := by
  by_cases hE : CompleteSpace E; swap; · simp [integral, hE, stronglyMeasurable_const]
  borelize E
  haveI : SeparableSpace (range (uncurry f) ∪ {0} : Set E) :=
    hf.separableSpace_range_union_singleton
  let s : ℕ → SimpleFunc (α × β) E :=
    SimpleFunc.approxOn _ hf.measurable (range (uncurry f) ∪ {0}) 0 (by simp)
  let s' : ℕ → α → SimpleFunc β E := fun n x => (s n).comp (Prod.mk x) measurable_prod_mk_left
  let f' : ℕ → α → E := fun n => {x | Integrable (f x) ν}.indicator fun x => (s' n x).integral ν
  have hf' : ∀ n, StronglyMeasurable (f' n) := by
    intro n; refine' StronglyMeasurable.indicator _ (measurableSet_integrable hf)
    have : ∀ x, ((s' n x).range.filter fun x => x ≠ 0) ⊆ (s n).range := by
      intro x; refine' Finset.Subset.trans (Finset.filter_subset _ _) _; intro y
      simp_rw [SimpleFunc.mem_range]; rintro ⟨z, rfl⟩; exact ⟨(x, z), rfl⟩
    simp only [SimpleFunc.integral_eq_sum_of_subset (this _)]
    refine' Finset.stronglyMeasurable_sum _ fun x _ => _
    refine' (Measurable.ennreal_toReal _).stronglyMeasurable.smul_const _
    simp (config := { singlePass := true }) only [SimpleFunc.coe_comp, preimage_comp]
    apply measurable_measure_prod_mk_left
    exact (s n).measurableSet_fiber x
  have h2f' : Tendsto f' atTop (𝓝 fun x : α => ∫ y : β, f x y ∂ν) := by
    rw [tendsto_pi_nhds]; intro x
    by_cases hfx : Integrable (f x) ν
    · have : ∀ n, Integrable (s' n x) ν := by
        intro n; apply (hfx.norm.add hfx.norm).mono' (s' n x).aestronglyMeasurable
        apply eventually_of_forall; intro y
        simp_rw [SimpleFunc.coe_comp]; exact SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n
      

Proof state:
case pos
α : Type u_1
α' : Type u_2
β : Type u_3
β' : Type u_4
γ : Type u_5
E : Type u_6
inst✝⁷ : MeasurableSpace α
inst✝⁶ : MeasurableSpace α'
inst✝⁵ : MeasurableSpace β
inst✝⁴ : MeasurableSpace β'
inst✝³ : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : SigmaFinite ν
f : α → β → E
hf : StronglyMeasurable (uncurry f)
hE : CompleteSpace E
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
this✝ : SeparableSpace ↑(range (uncurry f) ∪ {0})
s : ℕ → SimpleFunc (α × β) E :=
  SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) ∪ {0}) 0
    (_ : 0 ∈ range (uncurry f) ∪ {0})
s' : ℕ → α → SimpleFunc β E := fun n x => SimpleFunc.comp (s n) (Prod.mk x) (_ : Measurable (Prod.mk x))
f' : ℕ → α → E := fun n => indicator {x | Integrable (f x)} fun x => SimpleFunc.integral ν (s' n x)
hf' : ∀ (n : ℕ), StronglyMeasurable (f' n)
x : α
hfx : Integrable (f x)
this : ∀ (n : ℕ), Integrable ↑(s' n x)
⊢ Tendsto (fun i => f' i x) atTop (𝓝 (∫ (y : β), f x y ∂ν))

Next tactic:
simp only [hfx, SimpleFunc.integral_eq_integral _ (this _), indicator_of_mem,
        mem_setOf_eq]