signed measure

A signed measure on a measurable spaceMathworldPlanetmathPlanetmath (Ω,𝒮) is a function μ:𝒮{+} which is σ-additive ( and such that μ()=0.


  1. 1.

    The usual (positive) measureMathworldPlanetmath is a particular case of signed measure, in which |μ|=μ (see Jordan decomposition.)

  2. 2.

    Notice that the value - is not allowed. For some authors, a signed measure can only take finite values (so that + is not allowed either). This is sometimes useful because it turns the space of all signed measures into a normed vector spacePlanetmathPlanetmath, with the natural operations, and the norm given by μ=|μ|(Ω).

  3. 3.

    An important example of signed measures arises from the usual measures in the following way: Let (Ω,𝒮,μ) be a measure space, and let f be a (real valued) measurable functionMathworldPlanetmath such that


    Then a signed measure is defined by

Title signed measure
Canonical name SignedMeasure
Date of creation 2013-03-22 13:26:55
Last modified on 2013-03-22 13:26:55
Owner Koro (127)
Last modified by Koro (127)
Numerical id 8
Author Koro (127)
Entry type Definition
Classification msc 28A12