signed measure
A signed measure on a measurable space^{} $(\mathrm{\Omega},\mathcal{S})$ is a function $\mu :\mathcal{S}\to \mathbb{R}\cup \{+\mathrm{\infty}\}$ which is $\sigma $additive (http://planetmath.org/Additive) and such that $\mu (\mathrm{\varnothing})=0$.
Remarks.

1.
The usual (positive) measure^{} is a particular case of signed measure, in which $\mu =\mu $ (see Jordan decomposition.)

2.
Notice that the value $\mathrm{\infty}$ is not allowed. For some authors, a signed measure can only take finite values (so that $+\mathrm{\infty}$ is not allowed either). This is sometimes useful because it turns the space of all signed measures into a normed vector space^{}, with the natural operations, and the norm given by $\parallel \mu \parallel =\mu (\mathrm{\Omega})$.

3.
An important example of signed measures arises from the usual measures in the following way: Let $(\mathrm{\Omega},\mathcal{S},\mu )$ be a measure space, and let $f$ be a (real valued) measurable function^{} such that
$$ Then a signed measure is defined by
$$A\mapsto {\int}_{A}f\mathit{d}\mu .$$
Title  signed measure 

Canonical name  SignedMeasure 
Date of creation  20130322 13:26:55 
Last modified on  20130322 13:26:55 
Owner  Koro (127) 
Last modified by  Koro (127) 
Numerical id  8 
Author  Koro (127) 
Entry type  Definition 
Classification  msc 28A12 