/-- Lebesgue integral against the composition-product of two kernels. -/
theorem lintegral_compProd' (κ : kernel α β) [IsSFiniteKernel κ] (η : kernel (α × β) γ)
    [IsSFiniteKernel η] (a : α) {f : β → γ → ℝ≥0∞} (hf : Measurable (Function.uncurry f)) :
    ∫⁻ bc, f bc.1 bc.2 ∂(κ ⊗ₖ η) a = ∫⁻ b, ∫⁻ c, f b c ∂η (a, b) ∂κ a := by
  let F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
  have h : ∀ a, ⨆ n, F n a = Function.uncurry f a :=
    SimpleFunc.iSup_eapprox_apply (Function.uncurry f) hf
  simp only [Prod.forall, Function.uncurry_apply_pair] at h
  simp_rw [← h]
  have h_mono : Monotone F := fun i j hij b =>
    SimpleFunc.monotone_eapprox (Function.uncurry f) hij _
  rw [lintegral_iSup (fun n => (F n).measurable) h_mono]
  have : ∀ b, ∫⁻ c, ⨆ n, F n (b, c) ∂η (a, b) = ⨆ n, ∫⁻ c, F n (b, c) ∂η (a, b) := by
    intro a
    rw [lintegral_iSup]
    · exact fun n => (F n).measurable.comp measurable_prod_mk_left
    · exact fun i j hij b => h_mono hij _
  simp_rw [this]
  have h_some_meas_integral :
    ∀ f' : SimpleFunc (β × γ) ℝ≥0∞, Measurable fun b => ∫⁻ c, f' (b, c) ∂η (a, b) := by
    intro f'
    have :
      (fun b => ∫⁻ c, f' (b, c) ∂η (a, b)) =
        (fun ab => ∫⁻ c, f' (ab.2, c) ∂η ab) ∘ fun b => (a, b) :=
      by ext1 ab; rfl
    rw [this]
    refine' Measurable.comp _ measurable_prod_mk_left
    exact
      Measurable.lintegral_kernel_prod_right
        ((SimpleFunc.measurable _).comp (measurable_fst.snd.prod_mk measurable_snd))
  rw [lintegral_iSup]
  rotate_left
  · exact fun n => h_some_meas_integral (F n)
  · exact fun i j hij b => lintegral_mono fun c => h_mono hij _
  congr
  ext1 n
  -- Porting note: Added `(P := _)`
  refine' SimpleFunc.induction (P := fun f => (∫⁻ (a : β × γ), f a ∂(κ ⊗ₖ η) a =
      ∫⁻ (a_1 : β), ∫⁻ (c : γ), f (a_1, c) ∂η (a, a_1) ∂κ a)) _ _ (F n)
  · intro c s hs
    classical -- Porting note: Added `classical` for `Set.piecewise_eq_indicator`
    simp (config := { unfoldPartialApp := true }) only [SimpleFunc.const_zero,
      SimpleFunc.coe_piecewise, SimpleFunc.coe_const, SimpleFunc.coe_zero,
      Set.piecewise_eq_indicator, Function.const, lintegral_indicator_const hs]
    rw [compProd_apply κ η _ hs, ← lintegral_const_mul c _]
    swap
    · exact (measurable_kernel_prod_mk_left ((measurable_fst.snd.prod_mk measurable_snd) hs)).comp
        measurable_prod_mk_left
    congr
    ext1 b
    rw [lintegral_indicator_const_comp measurable_prod_mk_left hs]
    rfl
  · intro f f' _ hf_eq hf'_eq
    simp_rw [SimpleFunc.coe_add, Pi.add_apply]
    

Proof state:
case e_s.h.refine'_2
α : Type u_1
β : Type u_2
ι : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
s : Set (β × γ)
κ : ↥(kernel α β)
inst✝¹ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝ : IsSFiniteKernel η
a : α
f✝ : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f✝)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f✝)
h : ∀ (a : β) (b : γ), ⨆ n, ↑(SimpleFunc.eapprox (Function.uncurry f✝) n) (a, b) = f✝ a b
h_mono : Monotone F
this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, ↑(F n) (b, c) ∂η (a, b) = ⨆ n, ∫⁻ (c : γ), ↑(F n) (b, c) ∂η (a, b)
h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), ↑f' (b, c) ∂η (a, b)
n : ℕ
f f' : SimpleFunc (β × γ) ℝ≥0∞
a✝ : Disjoint (Function.support ↑f) (Function.support ↑f')
hf_eq : ∫⁻ (a : β × γ), ↑f a ∂(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) ∂η (a, a_1) ∂κ a
hf'_eq : ∫⁻ (a : β × γ), ↑f' a ∂(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f' (a_1, c) ∂η (a, a_1) ∂κ a
⊢ ∫⁻ (a : β × γ), ↑f a + ↑f' a ∂(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), ↑f (a_1, c) + ↑f' (a_1, c) ∂η (a, a_1) ∂κ a

Next tactic:
change
      ∫⁻ x, (f : β × γ → ℝ≥0∞) x + f' x ∂(κ ⊗ₖ η) a =
        ∫⁻ b, ∫⁻ c : γ, f (b, c) + f' (b, c) ∂η (a, b) ∂κ a