---
tag: WEBBT18D4X
kind: concept
name: Sigma-finite measure
msc: [28A12]
specializes: [PR11BZQQPG]
sources:
  - url: https://en.wikipedia.org/wiki/Σ-finite_measure
same_as:
  wikidata: Q291430
  wikipedia: Σ-finite measure
---

## Summary

A σ-finite measure is a measure for which the whole space is a countable union of sets of finite measure.
Lebesgue measure on $\R^n$ is the main example.
Many basic theorems, such as the Fubini–Tonelli theorem and the Radon–Nikodym theorem, assume σ-finiteness.

## Explanations

### L1 human

A σ-finite measure may give the whole space infinite size, but the space can be cut into countably many pieces of finite size.
The real line has infinite length, yet it is the union of the intervals $[-n, n]$, each of finite length.
Counting measure on the real line is not σ-finite: a set of finite counting measure is finite, and countably many finite sets cannot cover the line.