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Mathlib.Probability.Kernel.MeasureCompProd

Composition-Product of a measure and a kernel #

This operation, denoted by ⊗ₘ, takes μ : Measure α and κ : Kernel α β and creates μ ⊗ₘ κ : Measure (α × β). The integral of a function against μ ⊗ₘ κ is ∫⁻ x, f x ∂(μ ⊗ₘ κ) = ∫⁻ a, ∫⁻ b, f (a, b) ∂(κ a) ∂μ.

μ ⊗ₘ κ is defined as ((Kernel.const Unit μ) ⊗ₖ (Kernel.prodMkLeft Unit κ)) ().

Main definitions #

Notations #

noncomputable def MeasureTheory.Measure.compProd {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (μ : MeasureTheory.Measure α) (κ : ProbabilityTheory.Kernel α β) :

The composition-product of a measure and a kernel.

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    The composition-product of a measure and a kernel.

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    • One or more equations did not get rendered due to their size.
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      theorem MeasureTheory.Measure.compProd_of_not_sfinite {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (μ : MeasureTheory.Measure α) (κ : ProbabilityTheory.Kernel α β) (h : ¬MeasureTheory.SFinite μ) :
      μ.compProd κ = 0
      @[simp]
      theorem MeasureTheory.Measure.compProd_zero_right {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (μ : MeasureTheory.Measure α) :
      μ.compProd 0 = 0
      theorem MeasureTheory.Measure.compProd_apply {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] {s : Set (α × β)} (hs : MeasurableSet s) :
      (μ.compProd κ) s = ∫⁻ (a : α), (κ a) (Prod.mk a ⁻¹' s)μ
      @[simp]
      theorem MeasureTheory.Measure.compProd_apply_univ {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.IsMarkovKernel κ] :
      (μ.compProd κ) Set.univ = μ Set.univ
      theorem MeasureTheory.Measure.compProd_apply_prod {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] {s : Set α} {t : Set β} (hs : MeasurableSet s) (ht : MeasurableSet t) :
      (μ.compProd κ) (s ×ˢ t) = ∫⁻ (a : α) in s, (κ a) tμ
      theorem MeasureTheory.Measure.compProd_congr {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {κ η : ProbabilityTheory.Kernel α β} [ProbabilityTheory.IsSFiniteKernel κ] [ProbabilityTheory.IsSFiniteKernel η] (h : κ =ᵐ[μ] η) :
      μ.compProd κ = μ.compProd η
      theorem MeasureTheory.Measure.ae_compProd_of_ae_ae {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {κ : ProbabilityTheory.Kernel α β} {p : α × βProp} (hp : MeasurableSet {x : α × β | p x}) (h : ∀ᵐ (a : α) ∂μ, ∀ᵐ (b : β) ∂κ a, p (a, b)) :
      ∀ᵐ (x : α × β) ∂μ.compProd κ, p x
      theorem MeasureTheory.Measure.ae_ae_of_ae_compProd {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] {p : α × βProp} (h : ∀ᵐ (x : α × β) ∂μ.compProd κ, p x) :
      ∀ᵐ (a : α) ∂μ, ∀ᵐ (b : β) ∂κ a, p (a, b)
      theorem MeasureTheory.Measure.ae_compProd_iff {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] {p : α × βProp} (hp : MeasurableSet {x : α × β | p x}) :
      (∀ᵐ (x : α × β) ∂μ.compProd κ, p x) ∀ᵐ (a : α) ∂μ, ∀ᵐ (b : β) ∂κ a, p (a, b)
      @[simp]

      The composition product of a measure and a constant kernel is the product between the two measures.

      theorem MeasureTheory.Measure.compProd_add_left {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (μ ν : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν] (κ : ProbabilityTheory.Kernel α β) :
      (μ + ν).compProd κ = μ.compProd κ + ν.compProd κ
      theorem MeasureTheory.Measure.compProd_add_right {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (μ : MeasureTheory.Measure α) (κ η : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsSFiniteKernel κ] [ProbabilityTheory.IsSFiniteKernel η] :
      μ.compProd (κ + η) = μ.compProd κ + μ.compProd η
      @[simp]
      theorem MeasureTheory.Measure.fst_compProd {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} (μ : MeasureTheory.Measure α) [MeasureTheory.SFinite μ] (κ : ProbabilityTheory.Kernel α β) [ProbabilityTheory.IsMarkovKernel κ] :
      (μ.compProd κ).fst = μ
      theorem MeasureTheory.Measure.compProd_smul_left {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {κ : ProbabilityTheory.Kernel α β} (a : ENNReal) [MeasureTheory.SFinite μ] [ProbabilityTheory.IsSFiniteKernel κ] :
      (a μ).compProd κ = a μ.compProd κ
      theorem MeasureTheory.Measure.lintegral_compProd {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] {f : α × βENNReal} (hf : Measurable f) :
      ∫⁻ (x : α × β), f xμ.compProd κ = ∫⁻ (a : α), ∫⁻ (b : β), f (a, b)κ aμ
      theorem MeasureTheory.Measure.setLIntegral_compProd {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] {f : α × βENNReal} (hf : Measurable f) {s : Set α} (hs : MeasurableSet s) {t : Set β} (ht : MeasurableSet t) :
      ∫⁻ (x : α × β) in s ×ˢ t, f xμ.compProd κ = ∫⁻ (a : α) in s, ∫⁻ (b : β) in t, f (a, b)κ aμ
      @[deprecated MeasureTheory.Measure.setLIntegral_compProd]
      theorem MeasureTheory.Measure.set_lintegral_compProd {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] {f : α × βENNReal} (hf : Measurable f) {s : Set α} (hs : MeasurableSet s) {t : Set β} (ht : MeasurableSet t) :
      ∫⁻ (x : α × β) in s ×ˢ t, f xμ.compProd κ = ∫⁻ (a : α) in s, ∫⁻ (b : β) in t, f (a, b)κ aμ

      Alias of MeasureTheory.Measure.setLIntegral_compProd.

      theorem MeasureTheory.Measure.integrable_compProd_iff {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 × βE} (hf : MeasureTheory.AEStronglyMeasurable f (μ.compProd κ)) :
      MeasureTheory.Integrable f (μ.compProd κ) (∀ᵐ (x : α) ∂μ, MeasureTheory.Integrable (fun (y : β) => f (x, y)) (κ x)) MeasureTheory.Integrable (fun (x : α) => ∫ (y : β), f (x, y)κ x) μ
      theorem MeasureTheory.Measure.integral_compProd {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 E] {f : α × βE} (hf : MeasureTheory.Integrable f (μ.compProd κ)) :
      ∫ (x : α × β), f xμ.compProd κ = ∫ (a : α), ∫ (b : β), f (a, b)κ aμ
      theorem MeasureTheory.Measure.setIntegral_compProd {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 E] {s : Set α} (hs : MeasurableSet s) {t : Set β} (ht : MeasurableSet t) {f : α × βE} (hf : MeasureTheory.IntegrableOn f (s ×ˢ t) (μ.compProd κ)) :
      ∫ (x : α × β) in s ×ˢ t, f xμ.compProd κ = ∫ (a : α) in s, ∫ (b : β) in t, f (a, b)κ aμ
      @[deprecated MeasureTheory.Measure.setIntegral_compProd]
      theorem MeasureTheory.Measure.set_integral_compProd {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 E] {s : Set α} (hs : MeasurableSet s) {t : Set β} (ht : MeasurableSet t) {f : α × βE} (hf : MeasureTheory.IntegrableOn f (s ×ˢ t) (μ.compProd κ)) :
      ∫ (x : α × β) in s ×ˢ t, f xμ.compProd κ = ∫ (a : α) in s, ∫ (b : β) in t, f (a, b)κ aμ

      Alias of MeasureTheory.Measure.setIntegral_compProd.

      theorem MeasureTheory.Measure.dirac_compProd_apply {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) :
      ((MeasureTheory.Measure.dirac a).compProd κ) s = (κ a) (Prod.mk a ⁻¹' s)
      instance MeasureTheory.Measure.instSFiniteProdCompProd {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.compProd κ)
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      theorem MeasureTheory.Measure.absolutelyContinuous_compProd_left {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ ν : MeasureTheory.Measure α} [MeasureTheory.SFinite ν] (hμν : μ.AbsolutelyContinuous ν) (κ : ProbabilityTheory.Kernel α β) :
      (μ.compProd κ).AbsolutelyContinuous (ν.compProd κ)
      theorem MeasureTheory.Measure.absolutelyContinuous_compProd_right {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {κ η : ProbabilityTheory.Kernel α β} [MeasureTheory.SFinite μ] [ProbabilityTheory.IsSFiniteKernel η] (hκη : ∀ᵐ (a : α) ∂μ, (κ a).AbsolutelyContinuous (η a)) :
      (μ.compProd κ).AbsolutelyContinuous (μ.compProd η)
      theorem MeasureTheory.Measure.absolutelyContinuous_compProd {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ ν : MeasureTheory.Measure α} {κ η : ProbabilityTheory.Kernel α β} [MeasureTheory.SFinite ν] [ProbabilityTheory.IsSFiniteKernel η] (hμν : μ.AbsolutelyContinuous ν) (hκη : ∀ᵐ (a : α) ∂μ, (κ a).AbsolutelyContinuous (η a)) :
      (μ.compProd κ).AbsolutelyContinuous (ν.compProd η)
      theorem MeasureTheory.Measure.absolutelyContinuous_of_compProd {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ ν : MeasureTheory.Measure α} {κ η : ProbabilityTheory.Kernel α β} [MeasureTheory.SFinite μ] [ProbabilityTheory.IsSFiniteKernel κ] [h_zero : ∀ (a : α), NeZero (κ a)] (h : (μ.compProd κ).AbsolutelyContinuous (ν.compProd η)) :
      μ.AbsolutelyContinuous ν