In mathematics, a signed measure is a generalization of the concept of (positive) measure by allowing the set function to take negative values, i.e., to acquire sign.

There are two slightly different concepts of a signed measure, depending on whether or not one allows it to take infinite values. Signed measures are usually only allowed to take finite real values, while some textbooks allow them to take infinite values. To avoid confusion, this article will call these two cases "finite signed measures" and "extended signed measures".

Given a measurable space ${\displaystyle (X,\Sigma )}$ (that is, a set ${\displaystyle X}$ with a σ-algebra ${\displaystyle \Sigma }$ on it), an extended signed measure is a set function $${\displaystyle \mu :\Sigma \to \mathbb {R} \cup \{\infty ,-\infty \}}$$ such that ${\displaystyle \mu (\varnothing )=0}$ and ${\displaystyle \mu }$ is σ-additive – that is, it satisfies the equality $${\displaystyle \mu \left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu (A_{n})}$$ for any sequence ${\displaystyle A_{1},A_{2},\ldots ,A_{n},\ldots }$ of disjoint sets in ${\displaystyle \Sigma .}$ The series on the right must converge absolutely when the value of the left-hand side is finite. One consequence is that an extended signed measure can take ${\displaystyle +\infty }$ or ${\displaystyle -\infty }$ as a value, but not both. The expression ${\displaystyle \infty -\infty }$ is undefined and must be avoided.

A finite signed measure (a.k.a. real measure) is defined in the same way, except that it is only allowed to take real values. That is, it cannot take ${\displaystyle +\infty }$ or ${\displaystyle -\infty .}$

Finite signed measures form a real vector space, while extended signed measures do not because they are not closed under addition. On the other hand, measures are extended signed measures, but are not in general finite signed measures.

Consider a non-negative measure ${\displaystyle \nu }$ on the space (X, Σ) and a measurable function f: X → R such that

Then, a finite signed measure is given by

for all A in Σ.