Lebesgue decomposition theorem

Let μ and ν be two σ-finite signed measures in the measurable spaceMathworldPlanetmathPlanetmath (Ω,𝒮). There exist two σ-finite (http://planetmath.org/SigmaFinite) signed measures ν0 and ν1 such that:

  1. 1.


  2. 2.

    ν0μ (i.e. ν0 is absolutely continuousMathworldPlanetmath with respect to μ;)

  3. 3.

    ν1μ (i.e. ν1 and μ are singular.)

These two measuresMathworldPlanetmath are uniquely determined.

Title Lebesgue decomposition theorem
Canonical name LebesgueDecompositionTheorem
Date of creation 2013-03-22 13:26:28
Last modified on 2013-03-22 13:26:28
Owner Koro (127)
Last modified by Koro (127)
Numerical id 8
Author Koro (127)
Entry type Theorem
Classification msc 28A12