---
tag: VG2P7R91Z7
kind: concept
name: Signed measure
msc: [28A10]
sources:
  - url: https://en.wikipedia.org/wiki/Signed_measure
same_as:
  wikidata: Q1764371
  wikipedia: Signed measure
---

## Summary

A signed measure is a countably additive set function on a σ-algebra that may take negative values.
It models quantities such as electric charge, which can be positive in some regions and negative in others.
Every signed measure is the difference of two ordinary [[measures|PR11BZQQPG]], one of them finite.
Signed measures appear as differences of measures, as integrals $E \mapsto \int_E f \, d\mu$ of integrable functions, and as elements of dual spaces of function spaces.

## Explanations

### L1 human

An ordinary measure gives each set a size that is zero or positive.
A signed measure also allows negative sizes, like a bank account in which some regions bring money and others cost money.
It must still add up correctly over countably many disjoint pieces.
For example, on $[-1, 1]$ the rule $\nu(E) = \int_E x \, dx$ gives $\nu([0, 1]) = \tfrac12$ and $\nu([-1, 0]) = -\tfrac12$.