Signed measure

Builds on Measurable space Countable additivity Extended real number line

Definition

Let $(X, \Sigma)$ be a measurable space. A signed measure on $(X, \Sigma)$ is a map $\nu \colon \Sigma \to [-\infty, \infty]$ , with values in the extended real line, such that:

$\nu(\emptyset) = 0$ ;
$\nu$ takes at most one of the two values $+\infty$ and $-\infty$ ;
$\nu$ is countably additive: for pairwise disjoint $E_1, E_2, \ldots$ in $\Sigma$ , $\nu\Big( \bigcup_{n} E_n \Big) = \sum_{n} \nu(E_n),$ where the series converges absolutely when the left side is finite.

Remarks

Example: let $\mu$ be a measure on $(X, \Sigma)$ and $f \in L^1(\mu)$ real-valued. Then $\nu(E) = \int_E f \, d\mu$ is a signed measure with finite values. Countable additivity follows from the dominated convergence theorem, and absolute convergence from $\sum_n \abs{\nu(E_n)} \le \int \abs{f} \, d\mu$ . More generally, $\mu_1 - \mu_2$ is a signed measure whenever $\mu_1, \mu_2$ are measures and at least one of them is finite.

Non-example: on $X = \set{a, b}$ with all subsets measurable, no signed measure has $\nu(\set{a}) = +\infty$ and $\nu(\set{b}) = -\infty$ . Clause 2 excludes it, and indeed $\nu(\set{a, b}) = \nu(\set{a}) + \nu(\set{b})$ would be $\infty - \infty$ .

A finite signed measure is one with values in $\R$ . Every measure is a signed measure.

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Tag	`AE0VPNDDXZ`
URI	`m2://AE0VPNDDXZ`
Kind	definition
Writing	W0 stub
MSC	28A10
Defines	Signed measure
Uses	Measurable space, Countable additivity, Extended real number line
Changed	2 November 2026

Signed measure

Definition

Remarks

Reading path

Further reading

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