Theorems · Theorem · probability

MeasureTheory.Measure.integrable_compProd_iff

∀ {α : Type u_5} {β : Type u_6} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {κ : ProbabilityTheory.Kernel α β}
  {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] [ProbabilityTheory.IsSFiniteKernel κ] {E : Type u_8}
  [inst : NormedAddCommGroup E] {f : α × β → E},
  MeasureTheory.AEStronglyMeasurable f (μ.compProd κ) →
    (MeasureTheory.Integrable f (μ.compProd 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)

Defined in

Mathlib.Probability.Kernel.Composition.IntegralCompProd

Cited by

4 results in Mathlib

Foundations

Depth 258 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.SFiniteProbabilityTheory.IsSFiniteKernelNormedAddCommGroup