---
tag: 55RNFNN39R
kind: definition
name: Finite measure
msc: [28A12]
defines: 63Z1Q4FS23
uses: [P15VQS5DJ9, PR11BZQQPG, 059TJN0NZ1]
sources:
  - url: https://en.wikipedia.org/wiki/Finite_measure
---

## Definition

Let $(X, \Sigma, \mu)$ be a [[measure space|P15VQS5DJ9]].
The [[measure|PR11BZQQPG]] $\mu$ is *finite* if $\mu(X) < \infty$.

## Remarks

Lebesgue measure on the Borel sets of $[0, 1]$ is finite, since $\mu([0, 1]) = 1$.
Lebesgue measure on the Borel sets of $\R$ is not finite, since $\mu(\R) = \infty$.
If $\mu$ is finite, then $\mu(A) \le \mu(X) < \infty$ for every $A \in \Sigma$, by monotonicity.
Every [[probability measure|059TJN0NZ1]] is finite.
If $0 < \mu(X) < \infty$, then $\mu / \mu(X)$ is a probability measure.