Theorems · Theorem · probability

MeasureTheory.Measure.absolutelyContinuous_compProd_iff

∀ {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.SigmaFinite μ] [MeasureTheory.SigmaFinite ν]
  [ProbabilityTheory.IsSFiniteKernel κ] [ProbabilityTheory.IsSFiniteKernel η] [∀ (x : α), NeZero (κ x)],
  (μ.compProd κ).AbsolutelyContinuous (ν.compProd η) ↔
    μ.AbsolutelyContinuous ν ∧ (μ.compProd κ).AbsolutelyContinuous (μ.compProd η)

Defined in

Mathlib.Probability.Kernel.Composition.MeasureCompProd

Cited by

4 results in Mathlib

Foundations

Depth 233 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.SigmaFiniteMeasureTheory.SigmaFiniteProbabilityTheory.IsSFiniteKernelProbabilityTheory.IsSFiniteKernelNeZero