Theorems · Theorem · probability

ProbabilityTheory.integrable_compProd_iff

∀ {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 : NormedAddCommGroup E] {a : α} {κ : ProbabilityTheory.Kernel α β}
  [ProbabilityTheory.IsSFiniteKernel κ] {η : ProbabilityTheory.Kernel (α × β) γ} [ProbabilityTheory.IsSFiniteKernel η]
  ⦃f : β × γ → E⦄,
  MeasureTheory.AEStronglyMeasurable f ((κ.compProd η) a) →
    (MeasureTheory.Integrable f ((κ.compProd 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))

Defined in

Mathlib.Probability.Kernel.Composition.IntegralCompProd

Cited by

3 results in Mathlib

Foundations

Depth 257 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupProbabilityTheory.IsSFiniteKernelProbabilityTheory.IsSFiniteKernel