l3lab/ntp-mathlib huggingface dataset:
/-- 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
    

Proof state:
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✝¹ : MeasurableSpace E := borel E
this✝ : 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
⊢ Tendsto (fun i => f' i x) atTop (𝓝 (∫ (y : β), f x y ∂ν))

Next tactic:
by_cases hfx : Integrable (f x) ν
/-- 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
    · 

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✝¹ : MeasurableSpace E := borel E
this✝ : 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)
⊢ Tendsto (fun i => f' i x) atTop (𝓝 (∫ (y : β), f x y ∂ν))

Next tactic:
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
/-- 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
        

Proof state:
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✝¹ : MeasurableSpace E := borel E
this✝ : 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)
⊢ ∀ (n : ℕ), Integrable ↑(s' n x)

Next tactic:
intro n
/-- 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; 

Proof state:
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✝¹ : MeasurableSpace E := borel E
this✝ : 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)
n : ℕ
⊢ Integrable ↑(s' n x)

Next tactic:
apply (hfx.norm.add hfx.norm).mono' (s' n x).aestronglyMeasurable
/-- 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
        

Proof state:
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✝¹ : MeasurableSpace E := borel E
this✝ : 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)
n : ℕ
⊢ ∀ᵐ (a : β) ∂ν, ‖↑(s' n x) a‖ ≤ ((fun a => ‖f x a‖) + fun a => ‖f x a‖) a

Next tactic:
apply eventually_of_forall
/-- 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; 

Proof state:
case hp
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✝¹ : MeasurableSpace E := borel E
this✝ : 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)
n : ℕ
⊢ ∀ (x_1 : β), ‖↑(s' n x) x_1‖ ≤ ((fun a => ‖f x a‖) + fun a => ‖f x a‖) x_1

Next tactic:
intro y
/-- 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
        

Proof state:
case hp
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✝¹ : MeasurableSpace E := borel E
this✝ : 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)
n : ℕ
y : β
⊢ ‖↑(s' n x) y‖ ≤ ((fun a => ‖f x a‖) + fun a => ‖f x a‖) y

Next tactic:
simp_rw [SimpleFunc.coe_comp]
/-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
  Fubini's theorem is measurable. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right' [SigmaFinite ν] ⦃f : α × β → E⦄
    (hf : StronglyMeasurable f) : StronglyMeasurable fun x => ∫ y, f (x, y) ∂ν := by
  

Proof state:
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 f
⊢ StronglyMeasurable fun x => ∫ (y : β), f (x, y) ∂ν

Next tactic:
rw [← uncurry_curry f] at hf
/-- The Bochner integral is measurable. This shows that the integrand of (the right-hand-side of)
  Fubini's theorem is measurable. -/
theorem MeasureTheory.StronglyMeasurable.integral_prod_right' [SigmaFinite ν] ⦃f : α × β → E⦄
    (hf : StronglyMeasurable f) : StronglyMeasurable fun x => ∫ y, f (x, y) ∂ν := by
  rw [← uncurry_curry f] at hf; 

Proof state:
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 (curry f))
⊢ StronglyMeasurable fun x => ∫ (y : β), f (x, y) ∂ν

Next tactic:
exact hf.integral_prod_right
theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
    (h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ := by
  

Proof state:
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✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SigmaFinite ν
s : Set (α × β)
hs : MeasurableSet s
h2s : ↑↑(Measure.prod μ ν) s ≠ ⊤
⊢ Integrable fun x => ENNReal.toReal (↑↑ν (Prod.mk x ⁻¹' s))

Next tactic:
refine' ⟨(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, _⟩
theorem integrable_measure_prod_mk_left {s : Set (α × β)} (hs : MeasurableSet s)
    (h2s : (μ.prod ν) s ≠ ∞) : Integrable (fun x => (ν (Prod.mk x ⁻¹' s)).toReal) μ := by
  refine' ⟨(measurable_measure_prod_mk_left hs).ennreal_toReal.aemeasurable.aestronglyMeasurable, _⟩
  

Proof state:
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✝⁶ : MeasurableSpace α
inst✝⁵ : MeasurableSpace α'
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace β'
inst✝² : MeasurableSpace γ
μ μ' : Measure α
ν ν' : Measure β
τ : Measure γ
inst✝¹ : NormedAddCommGroup E
inst✝ : SigmaFinite ν
s : Set (α × β)
hs : MeasurableSet s
h2s : ↑↑(Measure.prod μ ν) s ≠ ⊤
⊢ HasFiniteIntegral fun x => ENNReal.toReal (↑↑ν (Prod.mk x ⁻¹' s))

Next tactic:
simp_rw [HasFiniteIntegral, ennnorm_eq_ofReal toReal_nonneg]