Theorems · Theorem · measure theory

MeasureTheory.Measure.eq_singularPart

∀ {α : Type u_1} {m : MeasurableSpace α} {μ ν s : MeasureTheory.Measure α} {f : α → ENNReal},
  Measurable f → s.MutuallySingular ν → μ = s + ν.withDensity f → s = μ.singularPart ν

Given measures μ and ν, if s is a measure mutually singular to ν and f is a measurable function such that μ = s + fν, then s = μ.singularPart μ. This theorem provides the uniqueness of the singularPart in the Lebesgue decomposition theorem, while MeasureTheory.Measure.eq_rnDeriv provides the uniqueness of the rnDeriv.

Defined in

Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Cited by

6 results in Mathlib

Foundations

Depth 210 from the axioms · uses propext, Classical.choice, Quot.sound