Theorems · Theorem · measure theory

MeasureTheory.Measure.haveLebesgueDecomposition_of_finiteMeasure

∀ {α : Type u_1} {m : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ]
  [MeasureTheory.IsFiniteMeasure ν], μ.HaveLebesgueDecomposition ν

Any pair of finite measures μ and ν, HaveLebesgueDecomposition. That is to say, there exist a measure ξ and a measurable function f, such that ξ is mutually singular with respect to ν and μ = ξ + ν.withDensity f. This is not an instance since this is also shown for the more general σ-finite measures with MeasureTheory.Measure.haveLebesgueDecomposition_of_sigmaFinite.

Defined in

Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue

Cited by

0 results in Mathlib

Foundations

Depth 209 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.IsFiniteMeasureMeasureTheory.IsFiniteMeasure