Theorems · Theorem · measure theory

MeasureTheory.conv_eq_withDensity_lconvolution_rnDeriv

∀ {G : Type u_3} [inst : AddGroup G] {mG : MeasurableSpace G} [MeasurableAdd₂ G] [MeasurableNeg G]
  {μ : MeasureTheory.Measure G} [μ.IsAddLeftInvariant] [MeasureTheory.SFinite μ] {ν₁ ν₂ : MeasureTheory.Measure G}
  [ν₁.HaveLebesgueDecomposition μ] [ν₂.HaveLebesgueDecomposition μ],
  ν₁.AbsolutelyContinuous μ →
    ν₂.AbsolutelyContinuous μ → ν₁.conv ν₂ = μ.withDensity (MeasureTheory.lconvolution (ν₁.rnDeriv μ) (ν₂.rnDeriv μ) μ)

Defined in

Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Cited by

3 results in Mathlib

Foundations

Depth 232 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddGroupMeasurableAdd₂MeasurableNegMeasureTheory.Measure.IsAddLeftInvariantMeasureTheory.SFiniteMeasureTheory.Measure.HaveLebesgueDecompositionMeasureTheory.Measure.HaveLebesgueDecomposition