/-- This is an auxiliary lemma for `measurable_kernel_prod_mk_left`. -/
theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : MeasurableSet t)
    (hκs : ∀ a, IsFiniteMeasure (κ a)) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) := by
  -- `t` is a measurable set in the product `α × β`: we use that the product σ-algebra is generated
  -- by boxes to prove the result by induction.
  -- Porting note: added motive
  refine' MeasurableSpace.induction_on_inter
    (C := fun t => Measurable fun a => κ a (Prod.mk a ⁻¹' t))
    generateFrom_prod.symm isPiSystem_prod _ _ _ _ ht
  ·-- case `t = ∅`
    simp only [preimage_empty, measure_empty, measurable_const]
  · -- case of a box: `t = t₁ ×ˢ t₂` for measurable sets `t₁` and `t₂`
    intro t' ht'
    simp only [Set.mem_image2, Set.mem_setOf_eq, exists_and_left] at ht'
    obtain ⟨t₁, ht₁, t₂, ht₂, rfl⟩ := ht'
    classical
    simp_rw [mk_preimage_prod_right_eq_if]
    have h_eq_ite : (fun a => κ a (ite (a ∈ t₁) t₂ ∅)) = fun a => ite (a ∈ t₁) (κ a t₂) 0 := by
      ext1 a
      split_ifs
      exacts [rfl, measure_empty]
    rw [h_eq_ite]
    exact Measurable.ite ht₁ (kernel.measurable_coe κ ht₂) measurable_const
  · -- we assume that the result is true for `t` and we prove it for `tᶜ`
    intro t' ht' h_meas
    have h_eq_sdiff : ∀ a, Prod.mk a ⁻¹' t'ᶜ = Set.univ \ Prod.mk a ⁻¹' t' := by
      intro a
      ext1 b
      simp only [mem_compl_iff, mem_preimage, mem_diff, mem_univ, true_and_iff]
    simp_rw [h_eq_sdiff]
    have :
      (fun a => κ a (Set.univ \ Prod.mk a ⁻¹' t')) = fun a =>
        κ a Set.univ - κ a (Prod.mk a ⁻¹' t') := by
      ext1 a
      rw [← Set.diff_inter_self_eq_diff, Set.inter_univ, measure_diff (Set.subset_univ _)]
      · exact (@measurable_prod_mk_left α β _ _ a) ht'
      · exact measure_ne_top _ _
    rw [this]
    exact Measurable.sub (kernel.measurable_coe κ MeasurableSet.univ) h_meas
  · -- we assume that the result is true for a family of disjoint sets and prove it for their union
    intro f h_disj hf_meas hf
    have h_Union :
      (fun a => κ a (Prod.mk a ⁻¹' ⋃ i, f i)) = fun a => κ a (⋃ i, Prod.mk a ⁻¹' f i) := by
      ext1 a
      congr with b
      simp only [mem_iUnion, mem_preimage]
    rw [h_Union]
    

Proof state:
case refine'_4
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
t : Set (α × β)
ht : MeasurableSet t
hκs : ∀ (a : α), IsFiniteMeasure (κ a)
f : ℕ → Set (α × β)
h_disj : Pairwise (Disjoint on f)
hf_meas : ∀ (i : ℕ), MeasurableSet (f i)
hf : ∀ (i : ℕ), (fun t => Measurable fun a => ↑↑(κ a) (Prod.mk a ⁻¹' t)) (f i)
h_Union : (fun a => ↑↑(κ a) (Prod.mk a ⁻¹' ⋃ i, f i)) = fun a => ↑↑(κ a) (⋃ i, Prod.mk a ⁻¹' f i)
⊢ Measurable fun a => ↑↑(κ a) (⋃ i, Prod.mk a ⁻¹' f i)

Next tactic:
have h_tsum :
      (fun a => κ a (⋃ i, Prod.mk a ⁻¹' f i)) = fun a => ∑' i, κ a (Prod.mk a ⁻¹' f i) := by
      ext1 a
      rw [measure_iUnion]
      · intro i j hij s hsi hsj b hbs
        have habi : {(a, b)} ⊆ f i := by rw [Set.singleton_subset_iff]; exact hsi hbs
        have habj : {(a, b)} ⊆ f j := by rw [Set.singleton_subset_iff]; exact hsj hbs
        simpa only [Set.bot_eq_empty, Set.le_eq_subset, Set.singleton_subset_iff,
          Set.mem_empty_iff_false] using h_disj hij habi habj
      · exact fun i => (@measurable_prod_mk_left α β _ _ a) (hf_meas i)