PlanetMath: signed measure

(more info)

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signed measure

(Definition)

A signed measure on a measurable space $(\Omega,\mathscr{S})$ is a function $\mu:\mathscr{S}\rightarrow \mathbb{R}\cup\{+\infty\}$ which is $\sigma$ -additive and such that $\mu(\emptyset)=0$ .

Remarks.

The usual (positive) measure is a particular case of signed measure, in which $\vert\mu\vert = \mu$ (see Jordan decomposition.)
Notice that the value $-\infty$ is not allowed. For some authors, a signed measure can only take finite values (so that $+\infty$ 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 $\Vert\mu\Vert = \vert\mu\vert(\Omega)$ .
An important example of signed measures arises from the usual measures in the following way: Let $(\Omega,\mathscr{S},\mu)$ be a measure space, and let be a (real valued) measurable function such that
$\displaystyle \int_{\{x\in \Omega:f(x)<0\}} \vert f\vert d\mu <\infty.$
Then a signed measure is defined by
$\displaystyle A\mapsto \int_A fd\mu.$