ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ProbabilityTheory.Kernel.rnDeriv_def'
lemma rnDeriv_def' (κ η : Kernel α γ) :
    rnDeriv κ η = fun a x ↦ ENNReal.ofReal (rnDerivAux κ (κ + η) a x)
      / ENNReal.ofReal (1 - rnDerivAux κ (κ + η) a x) 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ)
Goal: κ.rnDeriv η = fun a x => ENNReal.ofReal (κ.rnDerivAux (κ + η) a x) / ENNReal.ofReal (1 - κ.rnDerivAux (κ + η) a x)
-- ProbabilityTheory.Kernel.measurable_rnDeriv
lemma measurable_rnDeriv (κ η : Kernel α γ) :
    Measurable (fun p : α × γ ↦ rnDeriv κ η p.1 p.2) 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ)
Goal: Measurable fun p => κ.rnDeriv η p.1 p.2
-- ProbabilityTheory.Kernel.measurable_rnDeriv_right
lemma measurable_rnDeriv_right (κ η : Kernel α γ) (a : α) :
    Measurable (fun x : γ ↦ rnDeriv κ η a x) 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) (a : α)
Goal: Measurable fun x => κ.rnDeriv η a x
-- ProbabilityTheory.Kernel.rnDeriv_eq_top_iff
lemma rnDeriv_eq_top_iff (κ η : Kernel α γ) (a : α) (x : γ) :
    rnDeriv κ η a x = ∞ ↔ (a, x) ∈ mutuallySingularSet κ η 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) (a : α) (x : γ)
Goal: κ.rnDeriv η a x = ⊤ ↔ (a, x) ∈ κ.mutuallySingularSet η
-- ProbabilityTheory.Kernel.rnDeriv_eq_top_iff'
lemma rnDeriv_eq_top_iff' (κ η : Kernel α γ) (a : α) (x : γ) :
    rnDeriv κ η a x = ∞ ↔ x ∈ mutuallySingularSetSlice κ η a 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) (a : α) (x : γ)
Goal: κ.rnDeriv η a x = ⊤ ↔ x ∈ κ.mutuallySingularSetSlice η a
-- ProbabilityTheory.Kernel.singularPart_def
noncomputable
irreducible_def singularPart (κ η : Kernel α γ) [IsSFiniteKernel κ] [IsSFiniteKernel η] :
    Kernel α γ 
Arguments: {α : Type u_3} {γ : Type u_4} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsSFiniteKernel κ] [ProbabilityTheory.IsSFiniteKernel η]
Goal: κ.singularPart η =
  (κ + η).withDensity fun a x =>
    ↑(κ.rnDerivAux (κ + η) a x).toNNReal - ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal * κ.rnDeriv η a x
-- ProbabilityTheory.Kernel.measurable_singularPart_fun
lemma measurable_singularPart_fun (κ η : Kernel α γ) :
    Measurable (fun p : α × γ ↦ Real.toNNReal (rnDerivAux κ (κ + η) p.1 p.2)
      - Real.toNNReal (1 - rnDerivAux κ (κ + η) p.1 p.2) * rnDeriv κ η p.1 p.2) 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ)
Goal: Measurable fun p =>
  ↑(κ.rnDerivAux (κ + η) p.1 p.2).toNNReal - ↑(1 - κ.rnDerivAux (κ + η) p.1 p.2).toNNReal * κ.rnDeriv η p.1 p.2
-- ProbabilityTheory.Kernel.measurable_singularPart_fun_right
lemma measurable_singularPart_fun_right (κ η : Kernel α γ) (a : α) :
    Measurable (fun x : γ ↦ Real.toNNReal (rnDerivAux κ (κ + η) a x)
      - Real.toNNReal (1 - rnDerivAux κ (κ + η) a x) * rnDeriv κ η a x) 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) (a : α)
Goal: Measurable fun x => ↑(κ.rnDerivAux (κ + η) a x).toNNReal - ↑(1 - κ.rnDerivAux (κ + η) a x).toNNReal * κ.rnDeriv η a x
-- ProbabilityTheory.Kernel.singularPart_compl_mutuallySingularSetSlice
lemma singularPart_compl_mutuallySingularSetSlice (κ η : Kernel α γ) [IsSFiniteKernel κ]
    [IsSFiniteKernel η] (a : α) :
    singularPart κ η a (mutuallySingularSetSlice κ η a)ᶜ = 0 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsSFiniteKernel κ] [ProbabilityTheory.IsSFiniteKernel η] (a : α)
Goal: ((κ.singularPart η) a) (κ.mutuallySingularSetSlice η a)ᶜ = 0
-- ProbabilityTheory.Kernel.singularPart_of_subset_compl_mutuallySingularSetSlice
lemma singularPart_of_subset_compl_mutuallySingularSetSlice [IsFiniteKernel κ]
    [IsFiniteKernel η] {a : α} {s : Set γ} (hs : s ⊆ (mutuallySingularSetSlice κ η a)ᶜ) :
    singularPart κ η a s = 0 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} {κ η : ProbabilityTheory.Kernel α γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] {a : α} {s : Set γ} (hs : s ⊆ (κ.mutuallySingularSetSlice η a)ᶜ)
Goal: ((κ.singularPart η) a) s = 0
-- ProbabilityTheory.Kernel.singularPart_of_subset_mutuallySingularSetSlice
lemma singularPart_of_subset_mutuallySingularSetSlice [IsFiniteKernel κ]
    [IsFiniteKernel η] {a : α} {s : Set γ} (hsm : MeasurableSet s)
    (hs : s ⊆ mutuallySingularSetSlice κ η a) :
    singularPart κ η a s = κ a s 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} {κ η : ProbabilityTheory.Kernel α γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] {a : α} {s : Set γ} (hsm : MeasurableSet s) (hs : s ⊆ κ.mutuallySingularSetSlice η a)
Goal: ((κ.singularPart η) a) s = (κ a) s
-- ProbabilityTheory.Kernel.withDensity_rnDeriv_mutuallySingularSetSlice
lemma withDensity_rnDeriv_mutuallySingularSetSlice (κ η : Kernel α γ) [IsFiniteKernel κ]
    [IsFiniteKernel η] (a : α) :
    withDensity η (rnDeriv κ η) a (mutuallySingularSetSlice κ η a) = 0 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] (a : α)
Goal: ((η.withDensity (κ.rnDeriv η)) a) (κ.mutuallySingularSetSlice η a) = 0
-- ProbabilityTheory.Kernel.withDensity_rnDeriv_of_subset_mutuallySingularSetSlice
lemma withDensity_rnDeriv_of_subset_mutuallySingularSetSlice [IsFiniteKernel κ]
    [IsFiniteKernel η] {a : α} {s : Set γ}
    (hs : s ⊆ mutuallySingularSetSlice κ η a) :
    withDensity η (rnDeriv κ η) a s = 0 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} {κ η : ProbabilityTheory.Kernel α γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] {a : α} {s : Set γ} (hs : s ⊆ κ.mutuallySingularSetSlice η a)
Goal: ((η.withDensity (κ.rnDeriv η)) a) s = 0
-- ProbabilityTheory.Kernel.withDensity_rnDeriv_of_subset_compl_mutuallySingularSetSlice
lemma withDensity_rnDeriv_of_subset_compl_mutuallySingularSetSlice
    [IsFiniteKernel κ] [IsFiniteKernel η] {a : α} {s : Set γ} (hsm : MeasurableSet s)
    (hs : s ⊆ (mutuallySingularSetSlice κ η a)ᶜ) :
    withDensity η (rnDeriv κ η) a s = κ a s 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} {κ η : ProbabilityTheory.Kernel α γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] {a : α} {s : Set γ} (hsm : MeasurableSet s) (hs : s ⊆ (κ.mutuallySingularSetSlice η a)ᶜ)
Goal: ((η.withDensity (κ.rnDeriv η)) a) s = (κ a) s
-- ProbabilityTheory.Kernel.mutuallySingular_singularPart
lemma mutuallySingular_singularPart (κ η : Kernel α γ) [IsFiniteKernel κ] [IsFiniteKernel η]
    (a : α) :
    singularPart κ η a ⟂ₘ η a 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] (a : α)
Goal: (κ.singularPart η) a ⟂ₘ η a
-- ProbabilityTheory.Kernel.rnDeriv_add_singularPart
lemma rnDeriv_add_singularPart (κ η : Kernel α γ) [IsFiniteKernel κ] [IsFiniteKernel η] :
    withDensity η (rnDeriv κ η) + singularPart κ η = κ 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η]
Goal: η.withDensity (κ.rnDeriv η) + κ.singularPart η = κ
-- ProbabilityTheory.Kernel.singularPart_eq_zero_iff_apply_eq_zero
lemma singularPart_eq_zero_iff_apply_eq_zero (κ η : Kernel α γ) [IsFiniteKernel κ]
    [IsFiniteKernel η] (a : α) :
    singularPart κ η a = 0 ↔ singularPart κ η a (mutuallySingularSetSlice κ η a) = 0 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] (a : α)
Goal: (κ.singularPart η) a = 0 ↔ ((κ.singularPart η) a) (κ.mutuallySingularSetSlice η a) = 0
-- ProbabilityTheory.Kernel.withDensity_rnDeriv_eq_zero_iff_apply_eq_zero
lemma withDensity_rnDeriv_eq_zero_iff_apply_eq_zero (κ η : Kernel α γ) [IsFiniteKernel κ]
    [IsFiniteKernel η] (a : α) :
    withDensity η (rnDeriv κ η) a = 0
      ↔ withDensity η (rnDeriv κ η) a (mutuallySingularSetSlice κ η a)ᶜ = 0 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] (a : α)
Goal: (η.withDensity (κ.rnDeriv η)) a = 0 ↔ ((η.withDensity (κ.rnDeriv η)) a) (κ.mutuallySingularSetSlice η a)ᶜ = 0
-- ProbabilityTheory.Kernel.singularPart_eq_zero_iff_absolutelyContinuous
lemma singularPart_eq_zero_iff_absolutelyContinuous (κ η : Kernel α γ)
    [IsFiniteKernel κ] [IsFiniteKernel η] (a : α) :
    singularPart κ η a = 0 ↔ κ a ≪ η a 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] (a : α)
Goal: (κ.singularPart η) a = 0 ↔ κ a ≪ η a
-- ProbabilityTheory.Kernel.withDensity_rnDeriv_eq_zero_iff_mutuallySingular
lemma withDensity_rnDeriv_eq_zero_iff_mutuallySingular (κ η : Kernel α γ)
    [IsFiniteKernel κ] [IsFiniteKernel η] (a : α) :
    withDensity η (rnDeriv κ η) a = 0 ↔ κ a ⟂ₘ η a 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] (a : α)
Goal: (η.withDensity (κ.rnDeriv η)) a = 0 ↔ κ a ⟂ₘ η a
-- ProbabilityTheory.Kernel.singularPart_eq_zero_iff_measure_eq_zero
lemma singularPart_eq_zero_iff_measure_eq_zero (κ η : Kernel α γ)
    [IsFiniteKernel κ] [IsFiniteKernel η] (a : α) :
    singularPart κ η a = 0 ↔ κ a (mutuallySingularSetSlice κ η a) = 0 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] (a : α)
Goal: (κ.singularPart η) a = 0 ↔ (κ a) (κ.mutuallySingularSetSlice η a) = 0
-- ProbabilityTheory.Kernel.withDensity_rnDeriv_eq_zero_iff_measure_eq_zero
lemma withDensity_rnDeriv_eq_zero_iff_measure_eq_zero (κ η : Kernel α γ)
    [IsFiniteKernel κ] [IsFiniteKernel η] (a : α) :
    withDensity η (rnDeriv κ η) a = 0 ↔ κ a (mutuallySingularSetSlice κ η a)ᶜ = 0 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η] (a : α)
Goal: (η.withDensity (κ.rnDeriv η)) a = 0 ↔ (κ a) (κ.mutuallySingularSetSlice η a)ᶜ = 0
-- ProbabilityTheory.Kernel.measurableSet_absolutelyContinuous
@[measurability]
lemma measurableSet_absolutelyContinuous (κ η : Kernel α γ) [IsFiniteKernel κ] [IsFiniteKernel η] :
    MeasurableSet {a | κ a ≪ η a} 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η]
Goal: MeasurableSet {a | κ a ≪ η a}
-- ProbabilityTheory.Kernel.measurableSet_mutuallySingular
@[measurability]
lemma measurableSet_mutuallySingular (κ η : Kernel α γ) [IsFiniteKernel κ] [IsFiniteKernel η] :
    MeasurableSet {a | κ a ⟂ₘ η a} 
Arguments: {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ) [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η]
Goal: MeasurableSet {a | κ a ⟂ₘ η a}
-- ProbabilityTheory.hasFiniteIntegral_prod_mk_left
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) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] (a : α) {s : Set (β × γ)} (h2s : ((κ ⊗ₖ η) a) s ≠ ⊤)
Goal: MeasureTheory.HasFiniteIntegral (fun b => ((η (a, b)) (Prod.mk b ⁻¹' s)).toReal) (κ a)
-- ProbabilityTheory.integrable_kernel_prod_mk_left
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) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] (a : α) {s : Set (β × γ)} (hs : MeasurableSet s) (h2s : ((κ ⊗ₖ η) a) s ≠ ⊤)
Goal: MeasureTheory.Integrable (fun b => ((η (a, b)) (Prod.mk b ⁻¹' s)).toReal) (κ a)
-- MeasureTheory.AEStronglyMeasurable.integral_kernel_compProd
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) 
Arguments: {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} [NormedAddCommGroup E] {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {a : α} [NormedSpace ℝ E] ⦃f : β × γ → E⦄ (hf : MeasureTheory.AEStronglyMeasurable f ((κ ⊗ₖ η) a))
Goal: MeasureTheory.AEStronglyMeasurable (fun x => ∫ (y : γ), f (x, y) ∂η (a, x)) (κ a)
-- MeasureTheory.AEStronglyMeasurable.compProd_mk_left
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)) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {mγ : MeasurableSpace γ} {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {a : α} {δ : Type u_5} [TopologicalSpace δ] {f : β × γ → δ} (hf : MeasureTheory.AEStronglyMeasurable f ((κ ⊗ₖ η) a))
Goal: ∀ᵐ (x : β) ∂κ a, MeasureTheory.AEStronglyMeasurable (fun y => f (x, y)) (η (a, x))
-- ProbabilityTheory.hasFiniteIntegral_compProd_iff
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) 
Arguments: {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} [NormedAddCommGroup E] {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {a : α} ⦃f : β × γ → E⦄ (h1f : MeasureTheory.StronglyMeasurable f)
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)
-- ProbabilityTheory.hasFiniteIntegral_compProd_iff'
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) 
Arguments: {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} [NormedAddCommGroup E] {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {a : α} ⦃f : β × γ → E⦄ (h1f : MeasureTheory.AEStronglyMeasurable f ((κ ⊗ₖ η) 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)
-- ProbabilityTheory.integrable_compProd_iff
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) 
Arguments: {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} [NormedAddCommGroup E] {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {a : α} ⦃f : β × γ → E⦄ (hf : MeasureTheory.AEStronglyMeasurable f ((κ ⊗ₖ η) 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)
-- MeasureTheory.Integrable.compProd_mk_left_ae
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)) 
Arguments: {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} [NormedAddCommGroup E] {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {a : α} ⦃f : β × γ → E⦄ (hf : MeasureTheory.Integrable f ((κ ⊗ₖ η) a))
Goal: ∀ᵐ (x : β) ∂κ a, MeasureTheory.Integrable (fun y => f (x, y)) (η (a, x))
-- MeasureTheory.Integrable.integral_norm_compProd
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) 
Arguments: {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} [NormedAddCommGroup E] {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {a : α} ⦃f : β × γ → E⦄ (hf : MeasureTheory.Integrable f ((κ ⊗ₖ η) a))
Goal: MeasureTheory.Integrable (fun x => ∫ (y : γ), ‖f (x, y)‖ ∂η (a, x)) (κ a)
-- MeasureTheory.Integrable.integral_compProd
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) 
Arguments: {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} [NormedAddCommGroup E] {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {a : α} [NormedSpace ℝ E] ⦃f : β × γ → E⦄ (hf : MeasureTheory.Integrable f ((κ ⊗ₖ η) a))
Goal: MeasureTheory.Integrable (fun x => ∫ (y : γ), f (x, y) ∂η (a, x)) (κ a)
-- ProbabilityTheory.Kernel.integral_fn_integral_add
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 
Arguments: {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} [NormedAddCommGroup E] {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {a : α} [NormedSpace ℝ E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace ℝ E'] ⦃f g : β × γ → E⦄ (F : E → E') (hf : MeasureTheory.Integrable f ((κ ⊗ₖ η) a)) (hg : MeasureTheory.Integrable g ((κ ⊗ₖ η) 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
-- ProbabilityTheory.Kernel.integral_fn_integral_sub
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 
Arguments: {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} [NormedAddCommGroup E] {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {a : α} [NormedSpace ℝ E] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace ℝ E'] ⦃f g : β × γ → E⦄ (F : E → E') (hf : MeasureTheory.Integrable f ((κ ⊗ₖ η) a)) (hg : MeasureTheory.Integrable g ((κ ⊗ₖ η) 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
-- ProbabilityTheory.Kernel.lintegral_fn_integral_sub
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 
Arguments: {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} [NormedAddCommGroup E] {κ : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η] {a : α} [NormedSpace ℝ E] ⦃f g : β × γ → E⦄ (F : E → ℝ≥0∞) (hf : MeasureTheory.Integrable f ((κ ⊗ₖ η) a)) (hg : MeasureTheory.Integrable g ((κ ⊗ₖ η) 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