---
tag: YQ5C1SRE0W
kind: concept
name: Locally finite measure
msc: [28C15]
specializes: [DYR1QFNGNJ]
sources:
  - url: https://en.wikipedia.org/wiki/Locally_finite_measure
same_as:
  wikidata: Q2136937
  wikipedia: Locally finite measure
---

## Summary

A Borel measure on a topological space is locally finite if every point has an open neighbourhood of finite measure.
It may give infinite measure to the whole space, but never to a small region.
On locally compact Hausdorff spaces this is the same as being finite on compact sets.
Local finiteness is part of the definition of a Radon measure.

## Explanations

### L1 human

Locally finite means that, around each point, a small enough window holds only a finite amount of mass.
Length on the real line is locally finite: the whole line has infinite length, but each bounded interval has finite length.
Counting measure on the real line is not: every open interval holds infinitely many points.