-- Module: Mathlib.Probability.Kernel.MeasurableIntegral
Goal:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst : NormedAddCommGroup E
inst_1 : IsSFiniteKernel κ
inst_2 : NormedSpace ℝ E
f : α → β → E
hf : StronglyMeasurable (uncurry f)
pos :
  ∀ (hE : CompleteSpace E),
    StronglyMeasurable.{u_1, u_4} (β := E) fun (x : α) ↦
      @integral _ _ _ _ mβ ((κ : (a : α) → Measure β) x) fun (y : β) ↦ f x y
neg :
  ∀ (hE : ¬CompleteSpace E),
    StronglyMeasurable.{u_1, u_4} (β := E) fun (x : α) ↦
      @integral _ _ _ _ mβ ((κ : (a : α) → Measure β) x) fun (y : β) ↦ f x y
pos₁ :
  ∀ (hE : CompleteSpace E),
    StronglyMeasurable.{u_1, u_4} (β := E) fun (x : α) ↦
      @integral _ _ _ _ mβ ((κ : (a : α) → Measure β) x) fun (y : β) ↦ f x y
neg₁ neg₂ :
  ∀ (hE : ¬CompleteSpace E),
    StronglyMeasurable.{u_1, u_4} (β := E) fun (x : α) ↦
      @integral _ _ _ _ mβ ((κ : (a : α) → Measure β) x) fun (y : β) ↦ f x y
pos₂ pos₃ :
  ∀ (hE : CompleteSpace E),
    StronglyMeasurable.{u_1, u_4} (β := E) fun (x : α) ↦
      @integral _ _ _ _ mβ ((κ : (a : α) → Measure β) x) fun (y : β) ↦ f x y
pos₄ :
  ∀ (hE : CompleteSpace E),
    let this := borel E;
    BorelSpace E →
      StronglyMeasurable.{u_1, u_4} (β := E) fun (x : α) ↦
        @integral _ _ _ _ mβ ((κ : (a : α) → Measure β) x) fun (y : β) ↦ f x y
pos₅ pos₆ pos₇ pos₈ :
  ∀ (hE : CompleteSpace E),
    let this := borel E;
    BorelSpace E →
      ∀ (this : TopologicalSpace.SeparableSpace.{u_4} (↑(range (uncurry f) ∪ {0}) : Type u_4)),
        StronglyMeasurable.{u_1, u_4} (β := E) fun (x : α) ↦
          @integral _ _ _ _ mβ ((κ : (a : α) → Measure β) x) fun (y : β) ↦ f x y
pos₉ :
  ∀ (hE : CompleteSpace E),
    let this := borel E;
    ∀ (this_1 : BorelSpace E) (this_2 : TopologicalSpace.SeparableSpace.{u_4} (↑(range (uncurry f) ∪ {0}) : Type u_4)),
      let s := SimpleFunc.approxOn (uncurry f) ⋯ (range (uncurry f) ∪ {0}) 0 ⋯;
      let s' : ℕ → α → SimpleFunc β E := fun (n : ℕ) (x : α) ↦ (s n).comp (Prod.mk x) ⋯;
      let f' : ℕ → α → E := fun (n : ℕ) ↦
        indicator {x : α | @Integrable _ _ _ _ mβ (f x) ((κ : (a : α) → Measure β) x)} fun (x : α) ↦
          SimpleFunc.integral ((κ : (a : α) → Measure β) x) (s' n x);
      ∀ (hf' : ∀ (n : ℕ), StronglyMeasurable (f' n)),
        StronglyMeasurable.{u_1, u_4} (β := E) fun (x : α) ↦
          @integral _ _ _ _ mβ ((κ : (a : α) → Measure β) x) fun (y : β) ↦ f x y
pos₁₀ :
  ∀ (hE : CompleteSpace E),
    let this := borel E;
    ∀ (this_1 : BorelSpace E) (this_2 : TopologicalSpace.SeparableSpace.{u_4} (↑(range (uncurry f) ∪ {0}) : Type u_4)),
      let s := SimpleFunc.approxOn (uncurry f) ⋯ (range (uncurry f) ∪ {0}) 0 ⋯;
      let s' : ℕ → α → SimpleFunc β E := fun (n : ℕ) (x : α) ↦ (s n).comp (Prod.mk x) ⋯;
      let f' : ℕ → α → E := fun (n : ℕ) ↦
        indicator {x : α | @Integrable _ _ _ _ mβ (f x) ((κ : (a : α) → Measure β) x)} fun (x : α) ↦
          SimpleFunc.integral ((κ : (a : α) → Measure β) x) (s' n x);
      ∀ (hf' : ∀ (n : ℕ), StronglyMeasurable (f' n))
        (h2f' :
          Tendsto f' atTop (𝓝 fun (x : α) ↦ @integral _ _ _ _ mβ ((κ : (a : α) → Measure β) x) fun (y : β) ↦ f x y)),
        StronglyMeasurable.{u_1, u_4} (β := E) fun (x : α) ↦
          @integral _ _ _ _ mβ ((κ : (a : α) → Measure β) x) fun (y : β) ↦ f x y
⊢ StronglyMeasurable.{u_1, u_4} (β := E) fun (x : α) ↦
    @integral _ _ _ _ mβ ((κ : (a : α) → Measure β) x) fun (y : β) ↦ f x y

Tactic:
classical
  by_cases hE : CompleteSpace E; swap
  · simp [integral, hE, stronglyMeasurable_const]
  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_prodMk_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
      exact Finset.Subset.trans (Finset.filter_subset _ _) (SimpleFunc.range_comp_subset_range _ _)
    simp only [SimpleFunc.integral_eq_sum_of_subset (this _)]
    refine Finset.stronglyMeasurable_fun_sum _ fun x _ => ?_
    refine (Measurable.ennreal_toReal ?_).stronglyMeasurable.smul_const _
    simp only [s', SimpleFunc.coe_comp, preimage_comp]
    apply Kernel.measurable_kernel_prodMk_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
        apply (hfx.norm.add hfx.norm).mono' (s' n x).aestronglyMeasurable
        filter_upwards with y
        simp_rw [s', SimpleFunc.coe_comp]; exact SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n
      simp only [f', hfx, SimpleFunc.integral_eq_integral _ (this _), indicator_of_mem,
        mem_setOf_eq]
      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) ?_ ?_
      · -- Porting note: was
        -- exact fun n => Eventually.of_forall fun y =>
        --   SimpleFunc.norm_approxOn_zero_le _ _ (x, y) n
        exact fun n => Eventually.of_forall fun y =>
          SimpleFunc.norm_approxOn_zero_le hf.measurable (by simp) (x, y) n
      · refine Eventually.of_forall fun y => SimpleFunc.tendsto_approxOn hf.measurable (by simp) ?_
        apply subset_closure
        simp [-uncurry_apply_pair]
    · simp [f', hfx, integral_undef]
  exact stronglyMeasurable_of_tendsto _ hf' h2f'

Premises used: MeasureTheory.stronglyMeasurable_const, CompleteSpace, MeasureTheory.integral