Theorems · Theorem · measure theory

MeasureTheory.Measure.rnDeriv_eq_div

∀ {α : Type u_1} {m : MeasurableSpace α} {μ ν ξ : MeasureTheory.Measure α} [MeasureTheory.SigmaFinite μ]
  [MeasureTheory.SigmaFinite ν] [MeasureTheory.SigmaFinite ξ],
  μ.AbsolutelyContinuous ξ → ν.AbsolutelyContinuous ξ → μ.rnDeriv ν =ᵐ[ν] fun x => μ.rnDeriv ξ x / ν.rnDeriv ξ x

For any measure ξ dominating μ and ν, the Radon-Nikodym derivative of μ with respect to ν is ν-almost everywhere equal to the ratio of the Radon-Nikodym derivatives of μ and ν with respect to ξ.

Defined in

Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Cited by

0 results in Mathlib

Foundations

Depth 223 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.SigmaFiniteMeasureTheory.SigmaFiniteMeasureTheory.SigmaFinite