signed measure

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 (http://planetmath.org/Additive) and such that $\mu(\emptyset)=0$.

Remarks.

1. 1.

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

2. 2.

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 $\|\mu\|=|\mu|(\Omega)$.

3. 3.

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 $f$ be a (real valued) measurable function such that

 $\int_{\{x\in\Omega:f(x)<0\}}|f|d\mu<\infty.$

Then a signed measure is defined by

 $A\mapsto\int_{A}fd\mu.$
Title signed measure SignedMeasure 2013-03-22 13:26:55 2013-03-22 13:26:55 Koro (127) Koro (127) 8 Koro (127) Definition msc 28A12