ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.Measure.integrable_compProd_iff
lemma integrable_compProd_iff [SFinite μ] [IsSFiniteKernel κ] {E : Type*} [NormedAddCommGroup E]
    {f : α × β → E} (hf : AEStronglyMeasurable f (μ ⊗ₘ κ)) :
    Integrable f (μ ⊗ₘ κ) ↔
      (∀ᵐ x ∂μ, Integrable (fun y => f (x, y)) (κ x)) ∧
        Integrable (fun x => ∫ y, ‖f (x, y)‖ ∂(κ x)) μ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {κ : ProbabilityTheory.Kernel α β} [MeasureTheory.SFinite μ] [ProbabilityTheory.IsSFiniteKernel κ] {E : Type u_3} [NormedAddCommGroup E] {f : α × β → E} (hf : MeasureTheory.AEStronglyMeasurable f (μ ⊗ₘ κ))
Goal: MeasureTheory.Integrable f (μ ⊗ₘ κ) ↔
  (∀ᵐ (x : α) ∂μ, MeasureTheory.Integrable (fun y => f (x, y)) (κ x)) ∧
    MeasureTheory.Integrable (fun x => ∫ (y : β), ‖f (x, y)‖ ∂κ x) μ
-- MeasureTheory.Measure.integral_compProd
lemma integral_compProd [SFinite μ] [IsSFiniteKernel κ] {E : Type*}
    [NormedAddCommGroup E] [NormedSpace ℝ E]
    {f : α × β → E} (hf : Integrable f (μ ⊗ₘ κ)) :
    ∫ x, f x ∂(μ ⊗ₘ κ) = ∫ a, ∫ b, f (a, b) ∂(κ a) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {κ : ProbabilityTheory.Kernel α β} [MeasureTheory.SFinite μ] [ProbabilityTheory.IsSFiniteKernel κ] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {f : α × β → E} (hf : MeasureTheory.Integrable f (μ ⊗ₘ κ))
Goal: ∫ (x : α × β), f x ∂μ ⊗ₘ κ = ∫ (a : α), ∫ (b : β), f (a, b) ∂κ a ∂μ
-- MeasureTheory.Measure.setIntegral_compProd
lemma setIntegral_compProd [SFinite μ] [IsSFiniteKernel κ] {E : Type*}
    [NormedAddCommGroup E] [NormedSpace ℝ E]
    {s : Set α} (hs : MeasurableSet s) {t : Set β} (ht : MeasurableSet t)
    {f : α × β → E} (hf : IntegrableOn f (s ×ˢ t) (μ ⊗ₘ κ))  :
    ∫ x in s ×ˢ t, f x ∂(μ ⊗ₘ κ) = ∫ a in s, ∫ b in t, f (a, b) ∂(κ a) ∂μ 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {κ : ProbabilityTheory.Kernel α β} [MeasureTheory.SFinite μ] [ProbabilityTheory.IsSFiniteKernel κ] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set α} (hs : MeasurableSet s) {t : Set β} (ht : MeasurableSet t) {f : α × β → E} (hf : MeasureTheory.IntegrableOn f (s ×ˢ t) (μ ⊗ₘ κ))
Goal: ∫ (x : α × β) in s ×ˢ t, f x ∂μ ⊗ₘ κ = ∫ (a : α) in s, ∫ (b : β) in t, f (a, b) ∂κ a ∂μ
-- MeasureTheory.Measure.dirac_compProd_apply
lemma dirac_compProd_apply [MeasurableSingletonClass α] {a : α} [IsSFiniteKernel κ]
    {s : Set (α × β)} (hs : MeasurableSet s) :
    (Measure.dirac a ⊗ₘ κ) s = κ a (Prod.mk a ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α β} [MeasurableSingletonClass α] {a : α} [ProbabilityTheory.IsSFiniteKernel κ] {s : Set (α × β)} (hs : MeasurableSet s)
Goal: (MeasureTheory.Measure.dirac a ⊗ₘ κ) s = (κ a) (Prod.mk a ⁻¹' s)
-- MeasureTheory.Measure.dirac_unit_compProd
lemma dirac_unit_compProd (κ : Kernel Unit β) [IsSFiniteKernel κ] :
    Measure.dirac () ⊗ₘ κ = (κ ()).map (Prod.mk ()) 
Arguments: {β : Type u_2} {mβ : MeasurableSpace β} (κ : ProbabilityTheory.Kernel Unit β) [ProbabilityTheory.IsSFiniteKernel κ]
Goal: MeasureTheory.Measure.dirac () ⊗ₘ κ = MeasureTheory.Measure.map (Prod.mk ()) (κ ())
-- MeasureTheory.Measure.dirac_unit_compProd_const
lemma dirac_unit_compProd_const (μ : Measure β) [IsFiniteMeasure μ] :
    Measure.dirac () ⊗ₘ Kernel.const Unit μ = μ.map (Prod.mk ()) 
Arguments: {β : Type u_2} {mβ : MeasurableSpace β} (μ : MeasureTheory.Measure β) [MeasureTheory.IsFiniteMeasure μ]
Goal: MeasureTheory.Measure.dirac () ⊗ₘ ProbabilityTheory.Kernel.const Unit μ = MeasureTheory.Measure.map (Prod.mk ()) μ
-- MeasureTheory.Measure.snd_dirac_unit_compProd_const
@[simp]
lemma snd_dirac_unit_compProd_const (μ : Measure β) [IsFiniteMeasure μ] :
    snd (Measure.dirac () ⊗ₘ Kernel.const Unit μ) = μ 
Arguments: {β : Type u_2} {mβ : MeasurableSpace β} (μ : MeasureTheory.Measure β) [MeasureTheory.IsFiniteMeasure μ]
Goal: (MeasureTheory.Measure.dirac () ⊗ₘ ProbabilityTheory.Kernel.const Unit μ).snd = μ
-- ProbabilityTheory.Kernel.isRatCondKernelCDFAux_density_Iic
lemma isRatCondKernelCDFAux_density_Iic (κ : Kernel α (γ × ℝ)) [IsFiniteKernel κ] :
    IsRatCondKernelCDFAux (fun (p : α × γ) q ↦ density κ (fst κ) p.1 p.2 (Iic q)) κ (fst κ) 
Arguments: {α : Type u_1} {γ : Type u_3} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × ℝ)) [ProbabilityTheory.IsFiniteKernel κ]
Goal: ProbabilityTheory.IsRatCondKernelCDFAux (fun p q => κ.density κ.fst p.1 p.2 (Set.Iic ↑q)) κ κ.fst
-- ProbabilityTheory.Kernel.isRatCondKernelCDF_density_Iic
lemma isRatCondKernelCDF_density_Iic (κ : Kernel α (γ × ℝ)) [IsFiniteKernel κ] :
    IsRatCondKernelCDF (fun (p : α × γ) q ↦ density κ (fst κ) p.1 p.2 (Iic q)) κ (fst κ) 
Arguments: {α : Type u_1} {γ : Type u_3} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × ℝ)) [ProbabilityTheory.IsFiniteKernel κ]
Goal: ProbabilityTheory.IsRatCondKernelCDF (fun p q => κ.density κ.fst p.1 p.2 (Set.Iic ↑q)) κ κ.fst
-- ProbabilityTheory.Kernel.isCondKernelCDF_condKernelCDF
lemma isCondKernelCDF_condKernelCDF (κ : Kernel α (γ × ℝ)) [IsFiniteKernel κ] :
    IsCondKernelCDF (condKernelCDF κ) κ (fst κ) 
Arguments: {α : Type u_1} {γ : Type u_3} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × ℝ)) [ProbabilityTheory.IsFiniteKernel κ]
Goal: ProbabilityTheory.IsCondKernelCDF κ.condKernelCDF κ κ.fst
-- ProbabilityTheory.Kernel.compProd_fst_condKernelReal
lemma compProd_fst_condKernelReal (κ : Kernel α (γ × ℝ)) [IsFiniteKernel κ] :
    fst κ ⊗ₖ condKernelReal κ = κ 
Arguments: {α : Type u_1} {γ : Type u_3} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] (κ : ProbabilityTheory.Kernel α (γ × ℝ)) [ProbabilityTheory.IsFiniteKernel κ]
Goal: κ.fst ⊗ₖ κ.condKernelReal = κ
-- ProbabilityTheory.Kernel.compProd_fst_condKernelUnitReal
@[deprecated disintegrate (since := "2024-07-26")]
lemma compProd_fst_condKernelUnitReal (κ : Kernel Unit (α × ℝ)) [IsFiniteKernel κ] :
    fst κ ⊗ₖ condKernelUnitReal κ = κ 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (κ : ProbabilityTheory.Kernel Unit (α × ℝ)) [ProbabilityTheory.IsFiniteKernel κ]
Goal: κ.fst ⊗ₖ κ.condKernelUnitReal = κ
-- ProbabilityTheory.Kernel.borelMarkovFromReal_apply
lemma borelMarkovFromReal_apply (Ω : Type*) [Nonempty Ω] [MeasurableSpace Ω] [StandardBorelSpace Ω]
    (η : Kernel α ℝ) (a : α) :
    borelMarkovFromReal Ω η a
      = if η a (range (embeddingReal Ω))ᶜ = 0 then (η a).comap (embeddingReal Ω)
        else (Measure.dirac (range_nonempty (embeddingReal Ω)).choose).comap (embeddingReal Ω) 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (Ω : Type u_5) [Nonempty Ω] [MeasurableSpace Ω] [StandardBorelSpace Ω] (η : ProbabilityTheory.Kernel α ℝ) (a : α)
Goal: (ProbabilityTheory.Kernel.borelMarkovFromReal Ω η) a =
  if (η a) (Set.range (MeasureTheory.embeddingReal Ω))ᶜ = 0 then
    MeasureTheory.Measure.comap (MeasureTheory.embeddingReal Ω) (η a)
  else MeasureTheory.Measure.comap (MeasureTheory.embeddingReal Ω) (MeasureTheory.Measure.dirac (Exists.choose ⋯))
-- ProbabilityTheory.Kernel.borelMarkovFromReal_apply'
lemma borelMarkovFromReal_apply' (Ω : Type*) [Nonempty Ω] [MeasurableSpace Ω] [StandardBorelSpace Ω]
    (η : Kernel α ℝ) (a : α) {s : Set Ω} (hs : MeasurableSet s) :
    borelMarkovFromReal Ω η a s
      = if η a (range (embeddingReal Ω))ᶜ = 0 then η a (embeddingReal Ω '' s)
        else (embeddingReal Ω '' s).indicator 1 (range_nonempty (embeddingReal Ω)).choose 
Arguments: {α : Type u_1} {mα : MeasurableSpace α} (Ω : Type u_5) [Nonempty Ω] [MeasurableSpace Ω] [StandardBorelSpace Ω] (η : ProbabilityTheory.Kernel α ℝ) (a : α) {s : Set Ω} (hs : MeasurableSet s)
Goal: ((ProbabilityTheory.Kernel.borelMarkovFromReal Ω η) a) s =
  if (η a) (Set.range (MeasureTheory.embeddingReal Ω))ᶜ = 0 then (η a) (MeasureTheory.embeddingReal Ω '' s)
  else (MeasureTheory.embeddingReal Ω '' s).indicator 1 (Exists.choose ⋯)
-- ProbabilityTheory.Kernel.compProd_fst_borelMarkovFromReal_eq_comapRight_compProd
lemma compProd_fst_borelMarkovFromReal_eq_comapRight_compProd
    (κ : Kernel α (β × Ω)) [IsSFiniteKernel κ] (η : Kernel (α × β) ℝ) [IsSFiniteKernel η]
    (hη : (fst (map κ (Prod.map (id : β → β) (embeddingReal Ω))
        (measurable_id.prod_map (measurableEmbedding_embeddingReal Ω).measurable))) ⊗ₖ η
      = map κ (Prod.map (id : β → β) (embeddingReal Ω))
        (measurable_id.prod_map (measurableEmbedding_embeddingReal Ω).measurable)) :
    fst κ ⊗ₖ borelMarkovFromReal Ω η
      = comapRight (fst (map κ (Prod.map (id : β → β) (embeddingReal Ω))
          (measurable_id.prod_map (measurableEmbedding_embeddingReal Ω).measurable)) ⊗ₖ η)
        (MeasurableEmbedding.id.prod_mk (measurableEmbedding_embeddingReal Ω)) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_4} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] (κ : ProbabilityTheory.Kernel α (β × Ω)) [ProbabilityTheory.IsSFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) ℝ) [ProbabilityTheory.IsSFiniteKernel η] (hη :
  (κ.map (Prod.map id (MeasureTheory.embeddingReal Ω)) ⋯).fst ⊗ₖ η =
    κ.map (Prod.map id (MeasureTheory.embeddingReal Ω)) ⋯)
Goal: κ.fst ⊗ₖ ProbabilityTheory.Kernel.borelMarkovFromReal Ω η =
  ((κ.map (Prod.map id (MeasureTheory.embeddingReal Ω)) ⋯).fst ⊗ₖ η).comapRight ⋯
-- ProbabilityTheory.Kernel.compProd_fst_borelMarkovFromReal
lemma compProd_fst_borelMarkovFromReal (κ : Kernel α (β × Ω)) [IsSFiniteKernel κ]
    (η : Kernel (α × β) ℝ) [IsSFiniteKernel η]
    (hη : (fst (map κ (Prod.map (id : β → β) (embeddingReal Ω))
        (measurable_id.prod_map (measurableEmbedding_embeddingReal Ω).measurable))) ⊗ₖ η
      = map κ (Prod.map (id : β → β) (embeddingReal Ω))
        (measurable_id.prod_map (measurableEmbedding_embeddingReal Ω).measurable)) :
    fst κ ⊗ₖ borelMarkovFromReal Ω η = κ 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_4} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] (κ : ProbabilityTheory.Kernel α (β × Ω)) [ProbabilityTheory.IsSFiniteKernel κ] (η : ProbabilityTheory.Kernel (α × β) ℝ) [ProbabilityTheory.IsSFiniteKernel η] (hη :
  (κ.map (Prod.map id (MeasureTheory.embeddingReal Ω)) ⋯).fst ⊗ₖ η =
    κ.map (Prod.map id (MeasureTheory.embeddingReal Ω)) ⋯)
Goal: κ.fst ⊗ₖ ProbabilityTheory.Kernel.borelMarkovFromReal Ω η = κ
-- ProbabilityTheory.Kernel.compProd_fst_condKernelBorel
@[deprecated disintegrate (since := "2024-07-26")]
lemma compProd_fst_condKernelBorel (κ : Kernel α (γ × Ω)) [IsFiniteKernel κ] :
    fst κ ⊗ₖ condKernelBorel κ = κ 
Arguments: {α : Type u_1} {γ : Type u_3} {Ω : Type u_4} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ} [MeasurableSpace.CountablyGenerated γ] [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] (κ : ProbabilityTheory.Kernel α (γ × Ω)) [ProbabilityTheory.IsFiniteKernel κ]
Goal: κ.fst ⊗ₖ κ.condKernelBorel = κ
-- ProbabilityTheory.Kernel.compProd_fst_condKernelUnitBorel
@[deprecated disintegrate (since := "2024-07-26")]
lemma compProd_fst_condKernelUnitBorel (κ : Kernel Unit (α × Ω)) [IsFiniteKernel κ] :
    fst κ ⊗ₖ condKernelUnitBorel κ = κ 
Arguments: {α : Type u_1} {Ω : Type u_4} {mα : MeasurableSpace α} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] (κ : ProbabilityTheory.Kernel Unit (α × Ω)) [ProbabilityTheory.IsFiniteKernel κ]
Goal: κ.fst ⊗ₖ κ.condKernelUnitBorel = κ
-- MeasureTheory.Measure.condKernel_def
noncomputable
irreducible_def _root_.MeasureTheory.Measure.condKernel (ρ : Measure (α × Ω)) [IsFiniteMeasure ρ] :
    Kernel α Ω 
Arguments: {α : Type u_5} {Ω : Type u_6} {mα : MeasurableSpace α} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] (ρ : MeasureTheory.Measure (α × Ω)) [MeasureTheory.IsFiniteMeasure ρ]
Goal: ρ.condKernel = (ProbabilityTheory.Kernel.const Unit ρ).condKernelUnitBorel.comap (fun a => ((), a)) ⋯
-- MeasureTheory.Measure.condKernel_apply
lemma _root_.MeasureTheory.Measure.condKernel_apply (ρ : Measure (α × Ω)) [IsFiniteMeasure ρ]
    (a : α) :
    ρ.condKernel a = condKernelUnitBorel (const Unit ρ) ((), a) 
Arguments: {α : Type u_1} {Ω : Type u_4} {mα : MeasurableSpace α} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] (ρ : MeasureTheory.Measure (α × Ω)) [MeasureTheory.IsFiniteMeasure ρ] (a : α)
Goal: ρ.condKernel a = (ProbabilityTheory.Kernel.const Unit ρ).condKernelUnitBorel ((), a)
-- MeasureTheory.Measure.compProd_fst_condKernel
@[deprecated Measure.disintegrate (since := "2024-07-24")]
lemma _root_.MeasureTheory.Measure.compProd_fst_condKernel
    (ρ : Measure (α × Ω)) [IsFiniteMeasure ρ] :
    ρ.fst ⊗ₘ ρ.condKernel = ρ 
Arguments: {α : Type u_1} {Ω : Type u_4} {mα : MeasurableSpace α} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] (ρ : MeasureTheory.Measure (α × Ω)) [MeasureTheory.IsFiniteMeasure ρ]
Goal: ρ.fst ⊗ₘ ρ.condKernel = ρ
-- MeasureTheory.Measure.condKernel_apply_of_ne_zero_of_measurableSet
@[deprecated Measure.IsCondKernel.apply_of_ne_zero (since := "2024-07-24"), nolint unusedArguments]
lemma _root_.MeasureTheory.Measure.condKernel_apply_of_ne_zero_of_measurableSet
    [MeasurableSingletonClass α] {x : α} (hx : ρ.fst {x} ≠ 0) {s : Set Ω} (hs : MeasurableSet s) :
    ρ.condKernel x s = (ρ.fst {x})⁻¹ * ρ ({x} ×ˢ s) 
Arguments: {α : Type u_1} {Ω : Type u_4} {mα : MeasurableSpace α} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {ρ : MeasureTheory.Measure (α × Ω)} [MeasureTheory.IsFiniteMeasure ρ] [MeasurableSingletonClass α] {x : α} (hx : ρ.fst {x} ≠ 0) {s : Set Ω} (hs : MeasurableSet s)
Goal: (ρ.condKernel x) s = (ρ.fst {x})⁻¹ * ρ ({x} ×ˢ s)
-- MeasureTheory.Measure.condKernel_apply_of_ne_zero
lemma _root_.MeasureTheory.Measure.condKernel_apply_of_ne_zero [MeasurableSingletonClass α]
    {x : α} (hx : ρ.fst {x} ≠ 0) (s : Set Ω) :
    ρ.condKernel x s = (ρ.fst {x})⁻¹ * ρ ({x} ×ˢ s) 
Arguments: {α : Type u_1} {Ω : Type u_4} {mα : MeasurableSpace α} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] {ρ : MeasureTheory.Measure (α × Ω)} [MeasureTheory.IsFiniteMeasure ρ] [MeasurableSingletonClass α] {x : α} (hx : ρ.fst {x} ≠ 0) (s : Set Ω)
Goal: (ρ.condKernel x) s = (ρ.fst {x})⁻¹ * ρ ({x} ×ˢ s)
-- ProbabilityTheory.Kernel.compProd_fst_condKernelCountable
@[deprecated disintegrate (since := "2024-07-24")]
lemma compProd_fst_condKernelCountable (κ : Kernel α (β × Ω)) [IsFiniteKernel κ] :
    fst κ ⊗ₖ condKernelCountable (fun a ↦ (κ a).condKernel)
      (fun x y h ↦ by simp [apply_congr_of_mem_measurableAtom _ h]) = κ 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_4} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [Countable α] (κ : ProbabilityTheory.Kernel α (β × Ω)) [ProbabilityTheory.IsFiniteKernel κ]
Goal: κ.fst ⊗ₖ ProbabilityTheory.Kernel.condKernelCountable (fun a => (κ a).condKernel) ⋯ = κ
-- ProbabilityTheory.Kernel.condKernel_def
noncomputable
irreducible_def condKernel : Kernel (α × β) Ω 
Arguments: {α : Type u_5} {β : Type u_6} {Ω : Type u_7} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [h : MeasurableSpace.CountableOrCountablyGenerated α β] (κ : ProbabilityTheory.Kernel α (β × Ω)) [ProbabilityTheory.IsFiniteKernel κ]
Goal: κ.condKernel =
  if hα : Countable α then ProbabilityTheory.Kernel.condKernelCountable (fun a => (κ a).condKernel) ⋯
  else κ.condKernelBorel
-- ProbabilityTheory.Kernel.compProd_fst_condKernel
@[deprecated Kernel.disintegrate (since := "2024-07-26")]
lemma compProd_fst_condKernel : fst κ ⊗ₖ condKernel κ = κ 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_4} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [h : MeasurableSpace.CountableOrCountablyGenerated α β] (κ : ProbabilityTheory.Kernel α (β × Ω)) [ProbabilityTheory.IsFiniteKernel κ]
Goal: κ.fst ⊗ₖ κ.condKernel = κ
-- ProbabilityTheory.lintegral_condKernel_mem
lemma lintegral_condKernel_mem (a : α) {s : Set (β × Ω)} (hs : MeasurableSet s) :
    ∫⁻ x, Kernel.condKernel κ (a, x) {y | (x, y) ∈ s} ∂(Kernel.fst κ a) = κ a s 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [MeasurableSpace.CountableOrCountablyGenerated α β] {κ : ProbabilityTheory.Kernel α (β × Ω)} [ProbabilityTheory.IsFiniteKernel κ] (a : α) {s : Set (β × Ω)} (hs : MeasurableSet s)
Goal: ∫⁻ (x : β), (κ.condKernel (a, x)) {y | (x, y) ∈ s} ∂κ.fst a = (κ a) s
-- ProbabilityTheory.setLIntegral_condKernel_eq_measure_prod
lemma setLIntegral_condKernel_eq_measure_prod (a : α) {s : Set β} (hs : MeasurableSet s)
    {t : Set Ω} (ht : MeasurableSet t) :
    ∫⁻ b in s, Kernel.condKernel κ (a, b) t ∂(Kernel.fst κ a) = κ a (s ×ˢ t) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [MeasurableSpace.CountableOrCountablyGenerated α β] {κ : ProbabilityTheory.Kernel α (β × Ω)} [ProbabilityTheory.IsFiniteKernel κ] (a : α) {s : Set β} (hs : MeasurableSet s) {t : Set Ω} (ht : MeasurableSet t)
Goal: ∫⁻ (b : β) in s, (κ.condKernel (a, b)) t ∂κ.fst a = (κ a) (s ×ˢ t)
-- ProbabilityTheory.lintegral_condKernel
lemma lintegral_condKernel (hf : Measurable f) (a : α) :
    ∫⁻ b, ∫⁻ ω, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a) = ∫⁻ x, f x ∂(κ a) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [MeasurableSpace.CountableOrCountablyGenerated α β] {κ : ProbabilityTheory.Kernel α (β × Ω)} [ProbabilityTheory.IsFiniteKernel κ] {f : β × Ω → ℝ≥0∞} (hf : Measurable f) (a : α)
Goal: ∫⁻ (b : β), ∫⁻ (ω : Ω), f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫⁻ (x : β × Ω), f x ∂κ a
-- ProbabilityTheory.setLIntegral_condKernel
lemma setLIntegral_condKernel (hf : Measurable f) (a : α) {s : Set β}
    (hs : MeasurableSet s) {t : Set Ω} (ht : MeasurableSet t) :
    ∫⁻ b in s, ∫⁻ ω in t, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
      = ∫⁻ x in s ×ˢ t, f x ∂(κ a) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [MeasurableSpace.CountableOrCountablyGenerated α β] {κ : ProbabilityTheory.Kernel α (β × Ω)} [ProbabilityTheory.IsFiniteKernel κ] {f : β × Ω → ℝ≥0∞} (hf : Measurable f) (a : α) {s : Set β} (hs : MeasurableSet s) {t : Set Ω} (ht : MeasurableSet t)
Goal: ∫⁻ (b : β) in s, ∫⁻ (ω : Ω) in t, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫⁻ (x : β × Ω) in s ×ˢ t, f x ∂κ a
-- ProbabilityTheory.setLIntegral_condKernel_univ_right
lemma setLIntegral_condKernel_univ_right (hf : Measurable f) (a : α) {s : Set β}
    (hs : MeasurableSet s) :
    ∫⁻ b in s, ∫⁻ ω, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
      = ∫⁻ x in s ×ˢ Set.univ, f x ∂(κ a) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [MeasurableSpace.CountableOrCountablyGenerated α β] {κ : ProbabilityTheory.Kernel α (β × Ω)} [ProbabilityTheory.IsFiniteKernel κ] {f : β × Ω → ℝ≥0∞} (hf : Measurable f) (a : α) {s : Set β} (hs : MeasurableSet s)
Goal: ∫⁻ (b : β) in s, ∫⁻ (ω : Ω), f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫⁻ (x : β × Ω) in s ×ˢ Set.univ, f x ∂κ a
-- ProbabilityTheory.setLIntegral_condKernel_univ_left
lemma setLIntegral_condKernel_univ_left (hf : Measurable f) (a : α) {t : Set Ω}
    (ht : MeasurableSet t) :
    ∫⁻ b, ∫⁻ ω in t, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
      = ∫⁻ x in Set.univ ×ˢ t, f x ∂(κ a) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [MeasurableSpace.CountableOrCountablyGenerated α β] {κ : ProbabilityTheory.Kernel α (β × Ω)} [ProbabilityTheory.IsFiniteKernel κ] {f : β × Ω → ℝ≥0∞} (hf : Measurable f) (a : α) {t : Set Ω} (ht : MeasurableSet t)
Goal: ∫⁻ (b : β), ∫⁻ (ω : Ω) in t, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫⁻ (x : β × Ω) in Set.univ ×ˢ t, f x ∂κ a
-- MeasureTheory.AEStronglyMeasurable.integral_kernel_condKernel
lemma _root_.MeasureTheory.AEStronglyMeasurable.integral_kernel_condKernel (a : α)
    (hf : AEStronglyMeasurable f (κ a)) :
    AEStronglyMeasurable (fun x ↦ ∫ y, f (x, y) ∂(Kernel.condKernel κ (a, x)))
      (Kernel.fst κ a) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [MeasurableSpace.CountableOrCountablyGenerated α β] {κ : ProbabilityTheory.Kernel α (β × Ω)} [ProbabilityTheory.IsFiniteKernel κ] {E : Type u_4} {f : β × Ω → E} [NormedAddCommGroup E] [NormedSpace ℝ E] (a : α) (hf : MeasureTheory.AEStronglyMeasurable f (κ a))
Goal: MeasureTheory.AEStronglyMeasurable (fun x => ∫ (y : Ω), f (x, y) ∂κ.condKernel (a, x)) (κ.fst a)
-- ProbabilityTheory.integral_condKernel
lemma integral_condKernel (a : α) (hf : Integrable f (κ a)) :
    ∫ b, ∫ ω, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a) = ∫ x, f x ∂(κ a) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [MeasurableSpace.CountableOrCountablyGenerated α β] {κ : ProbabilityTheory.Kernel α (β × Ω)} [ProbabilityTheory.IsFiniteKernel κ] {E : Type u_4} {f : β × Ω → E} [NormedAddCommGroup E] [NormedSpace ℝ E] (a : α) (hf : MeasureTheory.Integrable f (κ a))
Goal: ∫ (b : β), ∫ (ω : Ω), f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫ (x : β × Ω), f x ∂κ a
-- ProbabilityTheory.setIntegral_condKernel
lemma setIntegral_condKernel (a : α) {s : Set β} (hs : MeasurableSet s)
    {t : Set Ω} (ht : MeasurableSet t) (hf : IntegrableOn f (s ×ˢ t) (κ a)) :
    ∫ b in s, ∫ ω in t, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
      = ∫ x in s ×ˢ t, f x ∂(κ a) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [MeasurableSpace.CountableOrCountablyGenerated α β] {κ : ProbabilityTheory.Kernel α (β × Ω)} [ProbabilityTheory.IsFiniteKernel κ] {E : Type u_4} {f : β × Ω → E} [NormedAddCommGroup E] [NormedSpace ℝ E] (a : α) {s : Set β} (hs : MeasurableSet s) {t : Set Ω} (ht : MeasurableSet t) (hf : MeasureTheory.IntegrableOn f (s ×ˢ t) (κ a))
Goal: ∫ (b : β) in s, ∫ (ω : Ω) in t, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫ (x : β × Ω) in s ×ˢ t, f x ∂κ a
-- ProbabilityTheory.setIntegral_condKernel_univ_right
lemma setIntegral_condKernel_univ_right (a : α) {s : Set β} (hs : MeasurableSet s)
    (hf : IntegrableOn f (s ×ˢ Set.univ) (κ a)) :
    ∫ b in s, ∫ ω, f (b, ω) ∂(Kernel.condKernel κ (a, b)) ∂(Kernel.fst κ a)
      = ∫ x in s ×ˢ Set.univ, f x ∂(κ a) 
Arguments: {α : Type u_1} {β : Type u_2} {Ω : Type u_3} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} [MeasurableSpace Ω] [StandardBorelSpace Ω] [Nonempty Ω] [MeasurableSpace.CountableOrCountablyGenerated α β] {κ : ProbabilityTheory.Kernel α (β × Ω)} [ProbabilityTheory.IsFiniteKernel κ] {E : Type u_4} {f : β × Ω → E} [NormedAddCommGroup E] [NormedSpace ℝ E] (a : α) {s : Set β} (hs : MeasurableSet s) (hf : MeasureTheory.IntegrableOn f (s ×ˢ Set.univ) (κ a))
Goal: ∫ (b : β) in s, ∫ (ω : Ω), f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫ (x : β × Ω) in s ×ˢ Set.univ, f x ∂κ a