theorem StronglyMeasurable.integral_kernel_prod_right ⦃f : α → β → E⦄
    (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun x => ∫ y, f x y ∂κ x := by
  classical
  borelize E
  haveI : TopologicalSpace.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) (κ x)}.indicator fun x => (s' n x).integral (κ x)
  have hf' : ∀ n, StronglyMeasurable (f' n) := by
    intro n; refine' StronglyMeasurable.indicator _ (measurableSet_kernel_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 kernel.measurable_kernel_prod_mk_left
    exact (s n).measurableSet_fiber x
  have h2f' : Tendsto f' atTop (𝓝 fun x : α => ∫ y : β, f x y ∂κ x) := by
    rw [tendsto_pi_nhds]; intro x
    by_cases hfx : Integrable (f x) (κ x)
    · have : ∀ n, Integrable (s' n x) (κ x) := 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
      simp only [ hfx, SimpleFunc.integral_eq_integral _ (this _), indicator_of_mem,
        mem_setOf_eq]
      

Proof state:
case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
κ : ↥(kernel α β)
η : ↥(kernel (α × β) γ)
a : α
E : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
f : α → β → E
hf : StronglyMeasurable (uncurry f)
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
this✝ : TopologicalSpace.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 (κ x) (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 =>
      ∫ (x_1 : β),
        ↑(SimpleFunc.comp
              (SimpleFunc.approxOn (uncurry f) (_ : Measurable (uncurry f)) (range (uncurry f) ∪ {0}) 0
                (_ : 0 ∈ range (uncurry f) ∪ {0}) i)
              (Prod.mk x) (_ : Measurable (Prod.mk x)))
          x_1 ∂κ x)
    atTop (𝓝 (∫ (y : β), f x y ∂κ x))

Next tactic:
refine'
        tendsto_integral_of_dominated_convergence (fun y => ‖f x y‖ + ‖f x y‖)
          (fun n => (s' n x).aestronglyMeasurable) (hfx.norm.add hfx.norm) _ _