Theorems · Theorem · probability

ProbabilityTheory.rnDeriv_compProd_withDensity_rnDeriv

∀ {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.IsFiniteMeasure μ] [MeasureTheory.IsFiniteMeasure ν]
  [ProbabilityTheory.IsFiniteKernel κ] [ProbabilityTheory.IsFiniteKernel η],
  ((ν.withDensity (μ.rnDeriv ν)).compProd κ).rnDeriv (ν.compProd η) =ᵐ[ν.compProd η]
    (μ.compProd κ).rnDeriv (ν.compProd η)

Auxiliary lemma for rnDeriv_measure_compProd_left.

Defined in

Mathlib.Probability.Kernel.Composition.RadonNikodym

Cited by

1 results in Mathlib

Foundations

Depth 230 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsFiniteMeasureMeasureTheory.IsFiniteMeasureProbabilityTheory.IsFiniteKernelProbabilityTheory.IsFiniteKernel