l3lab/ntp-mathlib huggingface dataset:
theorem kernel.integral_fn_integral_add ⦃f g : β × γ → E⦄ (F : E → E')
    (hf : Integrable f ((κ ⊗ₖ η) a)) (hg : Integrable g ((κ ⊗ₖ η) a)) :
    ∫ x, F (∫ y, f (x, y) + g (x, y) ∂η (a, x)) ∂κ a =
      ∫ x, F (∫ y, f (x, y) ∂η (a, x) + ∫ y, g (x, y) ∂η (a, x)) ∂κ a := by
  refine' integral_congr_ae _
  filter_upwards [hf.compProd_mk_left_ae, hg.compProd_mk_left_ae] with _ h2f h2g
  

Proof state:
case h
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
f g : β × γ → E
F : E → E'
hf : Integrable f
hg : Integrable g
a✝ : β
h2f : Integrable fun y => f (a✝, y)
h2g : Integrable fun y => g (a✝, y)
⊢ F (∫ (y : γ), f (a✝, y) + g (a✝, y) ∂η (a, a✝)) =
    F (∫ (y : γ), f (a✝, y) ∂η (a, a✝) + ∫ (y : γ), g (a✝, y) ∂η (a, a✝))

Next tactic:
simp [integral_add h2f h2g]
theorem kernel.integral_fn_integral_sub ⦃f g : β × γ → E⦄ (F : E → E')
    (hf : Integrable f ((κ ⊗ₖ η) a)) (hg : Integrable g ((κ ⊗ₖ η) a)) :
    ∫ x, F (∫ y, f (x, y) - g (x, y) ∂η (a, x)) ∂κ a =
      ∫ x, F (∫ y, f (x, y) ∂η (a, x) - ∫ y, g (x, y) ∂η (a, x)) ∂κ a := by
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
f g : β × γ → E
F : E → E'
hf : Integrable f
hg : Integrable g
⊢ ∫ (x : β), F (∫ (y : γ), f (x, y) - g (x, y) ∂η (a, x)) ∂κ a =
    ∫ (x : β), F (∫ (y : γ), f (x, y) ∂η (a, x) - ∫ (y : γ), g (x, y) ∂η (a, x)) ∂κ a

Next tactic:
refine' integral_congr_ae _
theorem kernel.integral_fn_integral_sub ⦃f g : β × γ → E⦄ (F : E → E')
    (hf : Integrable f ((κ ⊗ₖ η) a)) (hg : Integrable g ((κ ⊗ₖ η) a)) :
    ∫ x, F (∫ y, f (x, y) - g (x, y) ∂η (a, x)) ∂κ a =
      ∫ x, F (∫ y, f (x, y) ∂η (a, x) - ∫ y, g (x, y) ∂η (a, x)) ∂κ a := by
  refine' integral_congr_ae _
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
f g : β × γ → E
F : E → E'
hf : Integrable f
hg : Integrable g
⊢ (fun x => F (∫ (y : γ), f (x, y) - g (x, y) ∂η (a, x))) =ᵐ[κ a] fun x =>
    F (∫ (y : γ), f (x, y) ∂η (a, x) - ∫ (y : γ), g (x, y) ∂η (a, x))

Next tactic:
filter_upwards [hf.compProd_mk_left_ae, hg.compProd_mk_left_ae] with _ h2f h2g
theorem kernel.integral_fn_integral_sub ⦃f g : β × γ → E⦄ (F : E → E')
    (hf : Integrable f ((κ ⊗ₖ η) a)) (hg : Integrable g ((κ ⊗ₖ η) a)) :
    ∫ x, F (∫ y, f (x, y) - g (x, y) ∂η (a, x)) ∂κ a =
      ∫ x, F (∫ y, f (x, y) ∂η (a, x) - ∫ y, g (x, y) ∂η (a, x)) ∂κ a := by
  refine' integral_congr_ae _
  filter_upwards [hf.compProd_mk_left_ae, hg.compProd_mk_left_ae] with _ h2f h2g
  

Proof state:
case h
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
f g : β × γ → E
F : E → E'
hf : Integrable f
hg : Integrable g
a✝ : β
h2f : Integrable fun y => f (a✝, y)
h2g : Integrable fun y => g (a✝, y)
⊢ F (∫ (y : γ), f (a✝, y) - g (a✝, y) ∂η (a, a✝)) =
    F (∫ (y : γ), f (a✝, y) ∂η (a, a✝) - ∫ (y : γ), g (a✝, y) ∂η (a, a✝))

Next tactic:
simp [integral_sub h2f h2g]
theorem kernel.lintegral_fn_integral_sub ⦃f g : β × γ → E⦄ (F : E → ℝ≥0∞)
    (hf : Integrable f ((κ ⊗ₖ η) a)) (hg : Integrable g ((κ ⊗ₖ η) a)) :
    ∫⁻ x, F (∫ y, f (x, y) - g (x, y) ∂η (a, x)) ∂κ a =
      ∫⁻ x, F (∫ y, f (x, y) ∂η (a, x) - ∫ y, g (x, y) ∂η (a, x)) ∂κ a := by
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
f g : β × γ → E
F : E → ℝ≥0∞
hf : Integrable f
hg : Integrable g
⊢ ∫⁻ (x : β), F (∫ (y : γ), f (x, y) - g (x, y) ∂η (a, x)) ∂κ a =
    ∫⁻ (x : β), F (∫ (y : γ), f (x, y) ∂η (a, x) - ∫ (y : γ), g (x, y) ∂η (a, x)) ∂κ a

Next tactic:
refine' lintegral_congr_ae _
theorem kernel.lintegral_fn_integral_sub ⦃f g : β × γ → E⦄ (F : E → ℝ≥0∞)
    (hf : Integrable f ((κ ⊗ₖ η) a)) (hg : Integrable g ((κ ⊗ₖ η) a)) :
    ∫⁻ x, F (∫ y, f (x, y) - g (x, y) ∂η (a, x)) ∂κ a =
      ∫⁻ x, F (∫ y, f (x, y) ∂η (a, x) - ∫ y, g (x, y) ∂η (a, x)) ∂κ a := by
  refine' lintegral_congr_ae _
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
f g : β × γ → E
F : E → ℝ≥0∞
hf : Integrable f
hg : Integrable g
⊢ (fun x => F (∫ (y : γ), f (x, y) - g (x, y) ∂η (a, x))) =ᵐ[κ a] fun x =>
    F (∫ (y : γ), f (x, y) ∂η (a, x) - ∫ (y : γ), g (x, y) ∂η (a, x))

Next tactic:
filter_upwards [hf.compProd_mk_left_ae, hg.compProd_mk_left_ae] with _ h2f h2g
theorem kernel.lintegral_fn_integral_sub ⦃f g : β × γ → E⦄ (F : E → ℝ≥0∞)
    (hf : Integrable f ((κ ⊗ₖ η) a)) (hg : Integrable g ((κ ⊗ₖ η) a)) :
    ∫⁻ x, F (∫ y, f (x, y) - g (x, y) ∂η (a, x)) ∂κ a =
      ∫⁻ x, F (∫ y, f (x, y) ∂η (a, x) - ∫ y, g (x, y) ∂η (a, x)) ∂κ a := by
  refine' lintegral_congr_ae _
  filter_upwards [hf.compProd_mk_left_ae, hg.compProd_mk_left_ae] with _ h2f h2g
  

Proof state:
case h
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
f g : β × γ → E
F : E → ℝ≥0∞
hf : Integrable f
hg : Integrable g
a✝ : β
h2f : Integrable fun y => f (a✝, y)
h2g : Integrable fun y => g (a✝, y)
⊢ F (∫ (y : γ), f (a✝, y) - g (a✝, y) ∂η (a, a✝)) =
    F (∫ (y : γ), f (a✝, y) ∂η (a, a✝) - ∫ (y : γ), g (a✝, y) ∂η (a, a✝))

Next tactic:
simp [integral_sub h2f h2g]
theorem kernel.continuous_integral_integral :
    -- Continuous fun f : α × β →₁[(κ ⊗ₖ η) a] E => ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
    Continuous fun f : (MeasureTheory.Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))) =>
        ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
⊢ Continuous fun f => ∫ (x : β), ∫ (y : γ), ↑↑f (x, y) ∂η (a, x) ∂κ a

Next tactic:
rw [continuous_iff_continuousAt]
theorem kernel.continuous_integral_integral :
    -- Continuous fun f : α × β →₁[(κ ⊗ₖ η) a] E => ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
    Continuous fun f : (MeasureTheory.Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))) =>
        ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
  rw [continuous_iff_continuousAt]; 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
⊢ ∀ (x : ↥(Lp E 1)), ContinuousAt (fun f => ∫ (x : β), ∫ (y : γ), ↑↑f (x, y) ∂η (a, x) ∂κ a) x

Next tactic:
intro g
theorem kernel.continuous_integral_integral :
    -- Continuous fun f : α × β →₁[(κ ⊗ₖ η) a] E => ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
    Continuous fun f : (MeasureTheory.Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))) =>
        ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
  rw [continuous_iff_continuousAt]; intro g
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
g : ↥(Lp E 1)
⊢ ContinuousAt (fun f => ∫ (x : β), ∫ (y : γ), ↑↑f (x, y) ∂η (a, x) ∂κ a) g

Next tactic:
refine'
    tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_compProd
      (eventually_of_forall fun h => (L1.integrable_coeFn h).integral_compProd) _
theorem kernel.continuous_integral_integral :
    -- Continuous fun f : α × β →₁[(κ ⊗ₖ η) a] E => ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
    Continuous fun f : (MeasureTheory.Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))) =>
        ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
  rw [continuous_iff_continuousAt]; intro g
  refine'
    tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_compProd
      (eventually_of_forall fun h => (L1.integrable_coeFn h).integral_compProd) _
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
g : ↥(Lp E 1)
⊢ Tendsto (fun i => ∫⁻ (x : β), ↑‖∫ (y : γ), ↑↑i (x, y) ∂η (a, x) - ∫ (y : γ), ↑↑g (x, y) ∂η (a, x)‖₊ ∂κ a) (𝓝 g) (𝓝 0)

Next tactic:
simp_rw [←
    kernel.lintegral_fn_integral_sub (fun x => (‖x‖₊ : ℝ≥0∞)) (L1.integrable_coeFn _)
      (L1.integrable_coeFn g)]
theorem kernel.continuous_integral_integral :
    -- Continuous fun f : α × β →₁[(κ ⊗ₖ η) a] E => ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
    Continuous fun f : (MeasureTheory.Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))) =>
        ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
  rw [continuous_iff_continuousAt]; intro g
  refine'
    tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_compProd
      (eventually_of_forall fun h => (L1.integrable_coeFn h).integral_compProd) _
  simp_rw [←
    kernel.lintegral_fn_integral_sub (fun x => (‖x‖₊ : ℝ≥0∞)) (L1.integrable_coeFn _)
      (L1.integrable_coeFn g)]
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
g : ↥(Lp E 1)
⊢ Tendsto (fun i => ∫⁻ (x : β), ↑‖∫ (y : γ), ↑↑i (x, y) - ↑↑g (x, y) ∂η (a, x)‖₊ ∂κ a) (𝓝 g) (𝓝 0)

Next tactic:
refine' tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _ (fun i => zero_le _) _
theorem kernel.continuous_integral_integral :
    -- Continuous fun f : α × β →₁[(κ ⊗ₖ η) a] E => ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
    Continuous fun f : (MeasureTheory.Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))) =>
        ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
  rw [continuous_iff_continuousAt]; intro g
  refine'
    tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_compProd
      (eventually_of_forall fun h => (L1.integrable_coeFn h).integral_compProd) _
  simp_rw [←
    kernel.lintegral_fn_integral_sub (fun x => (‖x‖₊ : ℝ≥0∞)) (L1.integrable_coeFn _)
      (L1.integrable_coeFn g)]
  refine' tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _ (fun i => zero_le _) _
  · 

Proof state:
case refine'_1
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
g : ↥(Lp E 1)
⊢ ↥(Lp E 1) → ℝ≥0∞

Next tactic:
exact fun i => ∫⁻ x, ∫⁻ y, ‖i (x, y) - g (x, y)‖₊ ∂η (a, x) ∂κ a
theorem kernel.continuous_integral_integral :
    -- Continuous fun f : α × β →₁[(κ ⊗ₖ η) a] E => ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
    Continuous fun f : (MeasureTheory.Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))) =>
        ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
  rw [continuous_iff_continuousAt]; intro g
  refine'
    tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_compProd
      (eventually_of_forall fun h => (L1.integrable_coeFn h).integral_compProd) _
  simp_rw [←
    kernel.lintegral_fn_integral_sub (fun x => (‖x‖₊ : ℝ≥0∞)) (L1.integrable_coeFn _)
      (L1.integrable_coeFn g)]
  refine' tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _ (fun i => zero_le _) _
  · exact fun i => ∫⁻ x, ∫⁻ y, ‖i (x, y) - g (x, y)‖₊ ∂η (a, x) ∂κ a
  

Proof state:
case refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
g : ↥(Lp E 1)
⊢ Tendsto (fun i => ∫⁻ (x : β), ∫⁻ (y : γ), ↑‖↑↑i (x, y) - ↑↑g (x, y)‖₊ ∂η (a, x) ∂κ a) (𝓝 g) (𝓝 0)
case refine'_3
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
g : ↥(Lp E 1)
⊢ (fun i => ∫⁻ (x : β), ↑‖∫ (y : γ), ↑↑i (x, y) - ↑↑g (x, y) ∂η (a, x)‖₊ ∂κ a) ≤ fun i =>
    ∫⁻ (x : β), ∫⁻ (y : γ), ↑‖↑↑i (x, y) - ↑↑g (x, y)‖₊ ∂η (a, x) ∂κ a

Next tactic:
swap
theorem kernel.continuous_integral_integral :
    -- Continuous fun f : α × β →₁[(κ ⊗ₖ η) a] E => ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
    Continuous fun f : (MeasureTheory.Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))) =>
        ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
  rw [continuous_iff_continuousAt]; intro g
  refine'
    tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_compProd
      (eventually_of_forall fun h => (L1.integrable_coeFn h).integral_compProd) _
  simp_rw [←
    kernel.lintegral_fn_integral_sub (fun x => (‖x‖₊ : ℝ≥0∞)) (L1.integrable_coeFn _)
      (L1.integrable_coeFn g)]
  refine' tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _ (fun i => zero_le _) _
  · exact fun i => ∫⁻ x, ∫⁻ y, ‖i (x, y) - g (x, y)‖₊ ∂η (a, x) ∂κ a
  swap; · 

Proof state:
case refine'_3
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
g : ↥(Lp E 1)
⊢ (fun i => ∫⁻ (x : β), ↑‖∫ (y : γ), ↑↑i (x, y) - ↑↑g (x, y) ∂η (a, x)‖₊ ∂κ a) ≤ fun i =>
    ∫⁻ (x : β), ∫⁻ (y : γ), ↑‖↑↑i (x, y) - ↑↑g (x, y)‖₊ ∂η (a, x) ∂κ a

Next tactic:
exact fun i => lintegral_mono fun x => ennnorm_integral_le_lintegral_ennnorm _
theorem kernel.continuous_integral_integral :
    -- Continuous fun f : α × β →₁[(κ ⊗ₖ η) a] E => ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
    Continuous fun f : (MeasureTheory.Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))) =>
        ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
  rw [continuous_iff_continuousAt]; intro g
  refine'
    tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_compProd
      (eventually_of_forall fun h => (L1.integrable_coeFn h).integral_compProd) _
  simp_rw [←
    kernel.lintegral_fn_integral_sub (fun x => (‖x‖₊ : ℝ≥0∞)) (L1.integrable_coeFn _)
      (L1.integrable_coeFn g)]
  refine' tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _ (fun i => zero_le _) _
  · exact fun i => ∫⁻ x, ∫⁻ y, ‖i (x, y) - g (x, y)‖₊ ∂η (a, x) ∂κ a
  swap; · exact fun i => lintegral_mono fun x => ennnorm_integral_le_lintegral_ennnorm _
  

Proof state:
case refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
g : ↥(Lp E 1)
⊢ Tendsto (fun i => ∫⁻ (x : β), ∫⁻ (y : γ), ↑‖↑↑i (x, y) - ↑↑g (x, y)‖₊ ∂η (a, x) ∂κ a) (𝓝 g) (𝓝 0)

Next tactic:
show
    Tendsto
      (fun i : β × γ →₁[(κ ⊗ₖ η) a] E => ∫⁻ x, ∫⁻ y : γ, ‖i (x, y) - g (x, y)‖₊ ∂η (a, x) ∂κ a)
      (𝓝 g) (𝓝 0)
theorem kernel.continuous_integral_integral :
    -- Continuous fun f : α × β →₁[(κ ⊗ₖ η) a] E => ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
    Continuous fun f : (MeasureTheory.Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))) =>
        ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
  rw [continuous_iff_continuousAt]; intro g
  refine'
    tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_compProd
      (eventually_of_forall fun h => (L1.integrable_coeFn h).integral_compProd) _
  simp_rw [←
    kernel.lintegral_fn_integral_sub (fun x => (‖x‖₊ : ℝ≥0∞)) (L1.integrable_coeFn _)
      (L1.integrable_coeFn g)]
  refine' tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _ (fun i => zero_le _) _
  · exact fun i => ∫⁻ x, ∫⁻ y, ‖i (x, y) - g (x, y)‖₊ ∂η (a, x) ∂κ a
  swap; · exact fun i => lintegral_mono fun x => ennnorm_integral_le_lintegral_ennnorm _
  show
    Tendsto
      (fun i : β × γ →₁[(κ ⊗ₖ η) a] E => ∫⁻ x, ∫⁻ y : γ, ‖i (x, y) - g (x, y)‖₊ ∂η (a, x) ∂κ a)
      (𝓝 g) (𝓝 0)
  

Proof state:
case refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
g : ↥(Lp E 1)
⊢ Tendsto (fun i => ∫⁻ (x : β), ∫⁻ (y : γ), ↑‖↑↑i (x, y) - ↑↑g (x, y)‖₊ ∂η (a, x) ∂κ a) (𝓝 g) (𝓝 0)

Next tactic:
have : ∀ i : (MeasureTheory.Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))),
      Measurable fun z => (‖i z - g z‖₊ : ℝ≥0∞) := fun i =>
    ((Lp.stronglyMeasurable i).sub (Lp.stronglyMeasurable g)).ennnorm
theorem kernel.continuous_integral_integral :
    -- Continuous fun f : α × β →₁[(κ ⊗ₖ η) a] E => ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
    Continuous fun f : (MeasureTheory.Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))) =>
        ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
  rw [continuous_iff_continuousAt]; intro g
  refine'
    tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_compProd
      (eventually_of_forall fun h => (L1.integrable_coeFn h).integral_compProd) _
  simp_rw [←
    kernel.lintegral_fn_integral_sub (fun x => (‖x‖₊ : ℝ≥0∞)) (L1.integrable_coeFn _)
      (L1.integrable_coeFn g)]
  refine' tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _ (fun i => zero_le _) _
  · exact fun i => ∫⁻ x, ∫⁻ y, ‖i (x, y) - g (x, y)‖₊ ∂η (a, x) ∂κ a
  swap; · exact fun i => lintegral_mono fun x => ennnorm_integral_le_lintegral_ennnorm _
  show
    Tendsto
      (fun i : β × γ →₁[(κ ⊗ₖ η) a] E => ∫⁻ x, ∫⁻ y : γ, ‖i (x, y) - g (x, y)‖₊ ∂η (a, x) ∂κ a)
      (𝓝 g) (𝓝 0)
  have : ∀ i : (MeasureTheory.Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))),
      Measurable fun z => (‖i z - g z‖₊ : ℝ≥0∞) := fun i =>
    ((Lp.stronglyMeasurable i).sub (Lp.stronglyMeasurable g)).ennnorm
  

Proof state:
case refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
g : ↥(Lp E 1)
this : ∀ (i : ↥(Lp E 1)), Measurable fun z => ↑‖↑↑i z - ↑↑g z‖₊
⊢ Tendsto (fun i => ∫⁻ (x : β), ∫⁻ (y : γ), ↑‖↑↑i (x, y) - ↑↑g (x, y)‖₊ ∂η (a, x) ∂κ a) (𝓝 g) (𝓝 0)

Next tactic:
simp_rw [← kernel.lintegral_compProd _ _ _ (this _), ← L1.ofReal_norm_sub_eq_lintegral, ←
    ofReal_zero]
theorem kernel.continuous_integral_integral :
    -- Continuous fun f : α × β →₁[(κ ⊗ₖ η) a] E => ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
    Continuous fun f : (MeasureTheory.Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))) =>
        ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a := by
  rw [continuous_iff_continuousAt]; intro g
  refine'
    tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_compProd
      (eventually_of_forall fun h => (L1.integrable_coeFn h).integral_compProd) _
  simp_rw [←
    kernel.lintegral_fn_integral_sub (fun x => (‖x‖₊ : ℝ≥0∞)) (L1.integrable_coeFn _)
      (L1.integrable_coeFn g)]
  refine' tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds _ (fun i => zero_le _) _
  · exact fun i => ∫⁻ x, ∫⁻ y, ‖i (x, y) - g (x, y)‖₊ ∂η (a, x) ∂κ a
  swap; · exact fun i => lintegral_mono fun x => ennnorm_integral_le_lintegral_ennnorm _
  show
    Tendsto
      (fun i : β × γ →₁[(κ ⊗ₖ η) a] E => ∫⁻ x, ∫⁻ y : γ, ‖i (x, y) - g (x, y)‖₊ ∂η (a, x) ∂κ a)
      (𝓝 g) (𝓝 0)
  have : ∀ i : (MeasureTheory.Lp (α := β × γ) E 1 (((κ ⊗ₖ η) a) : Measure (β × γ))),
      Measurable fun z => (‖i z - g z‖₊ : ℝ≥0∞) := fun i =>
    ((Lp.stronglyMeasurable i).sub (Lp.stronglyMeasurable g)).ennnorm
  simp_rw [← kernel.lintegral_compProd _ _ _ (this _), ← L1.ofReal_norm_sub_eq_lintegral, ←
    ofReal_zero]
  

Proof state:
case refine'_2
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝⁷ : NormedAddCommGroup E
κ : ↥(kernel α β)
inst✝⁶ : IsSFiniteKernel κ
η : ↥(kernel (α × β) γ)
inst✝⁵ : IsSFiniteKernel η
a : α
inst✝⁴ : NormedSpace ℝ E
inst✝³ : CompleteSpace E
E' : Type u_5
inst✝² : NormedAddCommGroup E'
inst✝¹ : CompleteSpace E'
inst✝ : NormedSpace ℝ E'
g : ↥(Lp E 1)
this : ∀ (i : ↥(Lp E 1)), Measurable fun z => ↑‖↑↑i z - ↑↑g z‖₊
⊢ Tendsto (fun i => ENNReal.ofReal ‖i - g‖) (𝓝 g) (𝓝 (ENNReal.ofReal 0))

Next tactic:
refine' (continuous_ofReal.tendsto 0).comp _