Theorems · Theorem · measure theory

MeasureTheory.Measure.rnDeriv_withDensity_right_of_absolutelyContinuous

∀ {α : Type u_1} {m : MeasurableSpace α} {μ : MeasureTheory.Measure α} {f : α → ENNReal} {ν : MeasureTheory.Measure α}
  [μ.HaveLebesgueDecomposition ν] [MeasureTheory.SigmaFinite ν],
  μ.AbsolutelyContinuous ν →
    AEMeasurable f ν →
      (∀ᵐ (x : α) ∂ν, f x ≠ 0) →
        (∀ᵐ (x : α) ∂ν, f x ≠ ⊤) → μ.rnDeriv (ν.withDensity f) =ᵐ[ν] fun x => (f x)⁻¹ * μ.rnDeriv ν x

Auxiliary lemma for rnDeriv_withDensity_right.

Defined in

Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym

Cited by

1 results in Mathlib

Foundations

Depth 215 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.Measure.HaveLebesgueDecompositionMeasureTheory.SigmaFinite