File: Mathlib/Probability/Kernel/IntegralCompProd.lean

full_name: ProbabilityTheory.hasFiniteIntegral_prod_mk_left
code: theorem hasFiniteIntegral_prod_mk_left (a : α) {s : Set (β × γ)} (h2s : (κ ⊗ₖ η) a s ≠ ∞) :
    HasFiniteIntegral (fun b => (η (a, b) (Prod.mk b ⁻¹' s)).toReal) (κ a) 
goal: MeasureTheory.HasFiniteIntegral (fun b => ((η (a, b)) (Prod.mk b ⁻¹' s)).toReal) (κ a)

full_name: ProbabilityTheory.integrable_kernel_prod_mk_left
code: theorem integrable_kernel_prod_mk_left (a : α) {s : Set (β × γ)} (hs : MeasurableSet s)
    (h2s : (κ ⊗ₖ η) a s ≠ ∞) : Integrable (fun b => (η (a, b) (Prod.mk b ⁻¹' s)).toReal) (κ a) 
goal: MeasureTheory.Integrable (fun b => ((η (a, b)) (Prod.mk b ⁻¹' s)).toReal) (κ a)

full_name: MeasureTheory.AEStronglyMeasurable.integral_kernel_compProd
code: theorem _root_.MeasureTheory.AEStronglyMeasurable.integral_kernel_compProd [NormedSpace ℝ E]
    ⦃f : β × γ → E⦄ (hf : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
    AEStronglyMeasurable (fun x => ∫ y, f (x, y) ∂η (a, x)) (κ a) 
goal: MeasureTheory.AEStronglyMeasurable (fun x => ∫ (y : γ), f (x, y) ∂η (a, x)) (κ a)

full_name: MeasureTheory.AEStronglyMeasurable.compProd_mk_left
code: theorem _root_.MeasureTheory.AEStronglyMeasurable.compProd_mk_left {δ : Type*} [TopologicalSpace δ]
    {f : β × γ → δ} (hf : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
    ∀ᵐ x ∂κ a, AEStronglyMeasurable (fun y => f (x, y)) (η (a, x)) 
goal: ∀ᵐ (x : β) ∂κ a, MeasureTheory.AEStronglyMeasurable (fun y => f (x, y)) (η (a, x))

full_name: ProbabilityTheory.hasFiniteIntegral_compProd_iff
code: theorem hasFiniteIntegral_compProd_iff ⦃f : β × γ → E⦄ (h1f : StronglyMeasurable f) :
    HasFiniteIntegral f ((κ ⊗ₖ η) a) ↔
      (∀ᵐ x ∂κ a, HasFiniteIntegral (fun y => f (x, y)) (η (a, x))) ∧
        HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) 
goal: MeasureTheory.HasFiniteIntegral f ((κ ⊗ₖ η) a) ↔
  (∀ᵐ (x : β) ∂κ a, MeasureTheory.HasFiniteIntegral (fun y => f (x, y)) (η (a, x))) ∧
    MeasureTheory.HasFiniteIntegral (fun x => ∫ (y : γ), ‖f (x, y)‖ ∂η (a, x)) (κ a)

full_name: ProbabilityTheory.hasFiniteIntegral_compProd_iff'
code: theorem hasFiniteIntegral_compProd_iff' ⦃f : β × γ → E⦄
    (h1f : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
    HasFiniteIntegral f ((κ ⊗ₖ η) a) ↔
      (∀ᵐ x ∂κ a, HasFiniteIntegral (fun y => f (x, y)) (η (a, x))) ∧
        HasFiniteIntegral (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) 
goal: MeasureTheory.HasFiniteIntegral f ((κ ⊗ₖ η) a) ↔
  (∀ᵐ (x : β) ∂κ a, MeasureTheory.HasFiniteIntegral (fun y => f (x, y)) (η (a, x))) ∧
    MeasureTheory.HasFiniteIntegral (fun x => ∫ (y : γ), ‖f (x, y)‖ ∂η (a, x)) (κ a)

full_name: ProbabilityTheory.integrable_compProd_iff
code: theorem integrable_compProd_iff ⦃f : β × γ → E⦄ (hf : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
    Integrable f ((κ ⊗ₖ η) a) ↔
      (∀ᵐ x ∂κ a, Integrable (fun y => f (x, y)) (η (a, x))) ∧
        Integrable (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) 
goal: MeasureTheory.Integrable f ((κ ⊗ₖ η) a) ↔
  (∀ᵐ (x : β) ∂κ a, MeasureTheory.Integrable (fun y => f (x, y)) (η (a, x))) ∧
    MeasureTheory.Integrable (fun x => ∫ (y : γ), ‖f (x, y)‖ ∂η (a, x)) (κ a)

full_name: MeasureTheory.Integrable.compProd_mk_left_ae
code: theorem _root_.MeasureTheory.Integrable.compProd_mk_left_ae ⦃f : β × γ → E⦄
    (hf : Integrable f ((κ ⊗ₖ η) a)) : ∀ᵐ x ∂κ a, Integrable (fun y => f (x, y)) (η (a, x)) 
goal: ∀ᵐ (x : β) ∂κ a, MeasureTheory.Integrable (fun y => f (x, y)) (η (a, x))

full_name: MeasureTheory.Integrable.integral_norm_compProd
code: theorem _root_.MeasureTheory.Integrable.integral_norm_compProd ⦃f : β × γ → E⦄
    (hf : Integrable f ((κ ⊗ₖ η) a)) : Integrable (fun x => ∫ y, ‖f (x, y)‖ ∂η (a, x)) (κ a) 
goal: MeasureTheory.Integrable (fun x => ∫ (y : γ), ‖f (x, y)‖ ∂η (a, x)) (κ a)

full_name: MeasureTheory.Integrable.integral_compProd
code: theorem _root_.MeasureTheory.Integrable.integral_compProd [NormedSpace ℝ E]
    ⦃f : β × γ → E⦄ (hf : Integrable f ((κ ⊗ₖ η) a)) :
    Integrable (fun x => ∫ y, f (x, y) ∂η (a, x)) (κ a) 
goal: MeasureTheory.Integrable (fun x => ∫ (y : γ), f (x, y) ∂η (a, x)) (κ a)

full_name: ProbabilityTheory.Kernel.integral_fn_integral_add
code: 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 
goal: ∫ (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

full_name: ProbabilityTheory.Kernel.integral_fn_integral_sub
code: 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 
goal: ∫ (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

full_name: ProbabilityTheory.Kernel.lintegral_fn_integral_sub
code: 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 
goal: ∫⁻ (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

full_name: ProbabilityTheory.Kernel.integral_integral_add
code: theorem Kernel.integral_integral_add ⦃f g : β × γ → E⦄ (hf : Integrable f ((κ ⊗ₖ η) a))
    (hg : Integrable g ((κ ⊗ₖ η) a)) :
    ∫ x, ∫ y, f (x, y) + g (x, y) ∂η (a, x) ∂κ a =
      ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a + ∫ x, ∫ y, g (x, y) ∂η (a, x) ∂κ a 
goal: ∫ (x : β), ∫ (y : γ), f (x, y) + g (x, y) ∂η (a, x) ∂κ a =
  ∫ (x : β), ∫ (y : γ), f (x, y) ∂η (a, x) ∂κ a + ∫ (x : β), ∫ (y : γ), g (x, y) ∂η (a, x) ∂κ a

full_name: ProbabilityTheory.Kernel.integral_integral_add'
code: theorem Kernel.integral_integral_add' ⦃f g : β × γ → E⦄ (hf : Integrable f ((κ ⊗ₖ η) a))
    (hg : Integrable g ((κ ⊗ₖ η) a)) :
    ∫ x, ∫ y, (f + g) (x, y) ∂η (a, x) ∂κ a =
      ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a + ∫ x, ∫ y, g (x, y) ∂η (a, x) ∂κ a 
goal: ∫ (x : β), ∫ (y : γ), (f + g) (x, y) ∂η (a, x) ∂κ a =
  ∫ (x : β), ∫ (y : γ), f (x, y) ∂η (a, x) ∂κ a + ∫ (x : β), ∫ (y : γ), g (x, y) ∂η (a, x) ∂κ a

full_name: ProbabilityTheory.Kernel.integral_integral_sub
code: theorem Kernel.integral_integral_sub ⦃f g : β × γ → E⦄ (hf : Integrable f ((κ ⊗ₖ η) a))
    (hg : Integrable g ((κ ⊗ₖ η) a)) :
    ∫ x, ∫ y, f (x, y) - g (x, y) ∂η (a, x) ∂κ a =
      ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a - ∫ x, ∫ y, g (x, y) ∂η (a, x) ∂κ a 
goal: ∫ (x : β), ∫ (y : γ), f (x, y) - g (x, y) ∂η (a, x) ∂κ a =
  ∫ (x : β), ∫ (y : γ), f (x, y) ∂η (a, x) ∂κ a - ∫ (x : β), ∫ (y : γ), g (x, y) ∂η (a, x) ∂κ a

full_name: ProbabilityTheory.Kernel.integral_integral_sub'
code: theorem Kernel.integral_integral_sub' ⦃f g : β × γ → E⦄ (hf : Integrable f ((κ ⊗ₖ η) a))
    (hg : Integrable g ((κ ⊗ₖ η) a)) :
    ∫ x, ∫ y, (f - g) (x, y) ∂η (a, x) ∂κ a =
      ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a - ∫ x, ∫ y, g (x, y) ∂η (a, x) ∂κ a 
goal: ∫ (x : β), ∫ (y : γ), (f - g) (x, y) ∂η (a, x) ∂κ a =
  ∫ (x : β), ∫ (y : γ), f (x, y) ∂η (a, x) ∂κ a - ∫ (x : β), ∫ (y : γ), g (x, y) ∂η (a, x) ∂κ a

full_name: ProbabilityTheory.Kernel.continuous_integral_integral
code: 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 
goal: Continuous fun f => ∫ (x : β), ∫ (y : γ), ↑↑f (x, y) ∂η (a, x) ∂κ a

full_name: ProbabilityTheory.integral_compProd
code: theorem integral_compProd :
    ∀ {f : β × γ → E} (_ : Integrable f ((κ ⊗ₖ η) a)),
      ∫ z, f z ∂(κ ⊗ₖ η) a = ∫ x, ∫ y, f (x, y) ∂η (a, x) ∂κ a 
goal: MeasureTheory.Integrable f ((κ ⊗ₖ η) a) → ∫ (z : β × γ), f z ∂(κ ⊗ₖ η) a = ∫ (x : β), ∫ (y : γ), f (x, y) ∂η (a, x) ∂κ a

full_name: ProbabilityTheory.setIntegral_compProd
code: theorem setIntegral_compProd {f : β × γ → E} {s : Set β} {t : Set γ} (hs : MeasurableSet s)
    (ht : MeasurableSet t) (hf : IntegrableOn f (s ×ˢ t) ((κ ⊗ₖ η) a)) :
    ∫ z in s ×ˢ t, f z ∂(κ ⊗ₖ η) a = ∫ x in s, ∫ y in t, f (x, y) ∂η (a, x) ∂κ a 
goal: ∫ (z : β × γ) in s ×ˢ t, f z ∂(κ ⊗ₖ η) a = ∫ (x : β) in s, ∫ (y : γ) in t, f (x, y) ∂η (a, x) ∂κ a

full_name: ProbabilityTheory.setIntegral_compProd_univ_right
code: theorem setIntegral_compProd_univ_right (f : β × γ → E) {s : Set β} (hs : MeasurableSet s)
    (hf : IntegrableOn f (s ×ˢ univ) ((κ ⊗ₖ η) a)) :
    ∫ z in s ×ˢ univ, f z ∂(κ ⊗ₖ η) a = ∫ x in s, ∫ y, f (x, y) ∂η (a, x) ∂κ a 
goal: ∫ (z : β × γ) in s ×ˢ Set.univ, f z ∂(κ ⊗ₖ η) a = ∫ (x : β) in s, ∫ (y : γ), f (x, y) ∂η (a, x) ∂κ a

full_name: ProbabilityTheory.setIntegral_compProd_univ_left
code: theorem setIntegral_compProd_univ_left (f : β × γ → E) {t : Set γ} (ht : MeasurableSet t)
    (hf : IntegrableOn f (univ ×ˢ t) ((κ ⊗ₖ η) a)) :
    ∫ z in univ ×ˢ t, f z ∂(κ ⊗ₖ η) a = ∫ x, ∫ y in t, f (x, y) ∂η (a, x) ∂κ a 
goal: ∫ (z : β × γ) in Set.univ ×ˢ t, f z ∂(κ ⊗ₖ η) a = ∫ (x : β), ∫ (y : γ) in t, f (x, y) ∂η (a, x) ∂κ a