Theorems · Theorem · measure theory

MeasurableEmbedding.map_withDensity_rnDeriv

∀ {α : Type u_1} {β : Type u_2} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {f : α → β},
  MeasurableEmbedding f →
    ∀ (μ ν : MeasureTheory.Measure α) [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν],
      MeasureTheory.Measure.map f (ν.withDensity (μ.rnDeriv ν)) =
        (MeasureTheory.Measure.map f ν).withDensity
          ((MeasureTheory.Measure.map f μ).rnDeriv (MeasureTheory.Measure.map f ν))

Defined in

Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Cited by

1 results in Mathlib

Foundations

Depth 218 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.SigmaFiniteMeasureTheory.SigmaFinite