Theorems · Theorem · probability

ProbabilityTheory.Kernel.rnDeriv_add_singularPart

∀ {α : Type u_1} {γ : Type u_2} {mα : MeasurableSpace α} {mγ : MeasurableSpace γ}
  [hαγ : MeasurableSpace.CountableOrCountablyGenerated α γ] (κ η : ProbabilityTheory.Kernel α γ)
  [inst : ProbabilityTheory.IsFiniteKernel κ] [inst_1 : ProbabilityTheory.IsFiniteKernel η],
  η.withDensity (κ.rnDeriv η) + κ.singularPart η = κ

Lebesgue decomposition of a finite kernel κ with respect to another one η. κ is the sum of an absolutely continuous part withDensity η (rnDeriv κ η) and a singular part singularPart κ η.

Defined in

Mathlib.Probability.Kernel.RadonNikodym

Cited by

7 results in Mathlib

Foundations

Depth 332 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasurableSpace.CountableOrCountablyGeneratedProbabilityTheory.IsFiniteKernelProbabilityTheory.IsFiniteKernel