Hahn decomposition theorem

Let μ be a signed measure in the measurable spaceMathworldPlanetmathPlanetmath (Ω,𝒮). There are two measurable sets A and B such that:

  1. 1.

    AB=Ω and AB=;

  2. 2.

    μ(E)0 for each E𝒮 such that EA;

  3. 3.

    μ(E)0 for each E𝒮 such that EB.

The pair (A,B) is called a Hahn decomposition for μ. This decomposition is not unique, but any other such decomposition (A,B) satisfies μ(AA)=μ(BB)=0 (where denotes the symmetric differenceMathworldPlanetmathPlanetmath), so the two decompositions differ in a set of measureMathworldPlanetmathPlanetmath 0.

Title Hahn decomposition theorem
Canonical name HahnDecompositionTheorem
Date of creation 2013-03-22 13:26:59
Last modified on 2013-03-22 13:26:59
Owner Koro (127)
Last modified by Koro (127)
Numerical id 10
Author Koro (127)
Entry type Theorem
Classification msc 28A12
Defines Hahn decomposition