A signed measure on a measurable space is a function which is $\sigma$ -additive and such that $\mu(\emptyset)=0$ .

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 $-\infty$ 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. An important example of signed measures arises from the usual measures in the following way: Let 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 $$A\mapsto \int_A fd\mu.$$