signed measure

A signed measure on a measurable space $(\mathrm{\Omega },𝒮)$ is a function $\mu :𝒮\to ℝ\cup \{+\mathrm{\infty }\}$ 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 $(\mathrm{\Omega },𝒮,\mu )$ be a measure space, and let $f$ be a (real valued) measurable function such that

   Then a signed measure is defined by

   $$A\mapsto {\int }_{A}f𝑑\mu .$$