---
tag: HZNEQQBCW4
kind: concept
name: Radon measure
msc: [28C15, 28C05]
specializes: [YQ5C1SRE0W, 28E5Q5BM69]
sources:
  - url: https://en.wikipedia.org/wiki/Radon_measure
same_as:
  wikidata: Q2126650
  wikipedia: Radon measure
---

## Summary

A Radon measure on a Hausdorff topological space is a Borel measure that is [[locally finite|YQ5C1SRE0W]] and [[inner regular|28E5Q5BM69]] with respect to compact sets.
Radon measures are the measures that fit well with the topology.
On a locally compact Hausdorff space, the Riesz–Markov–Kakutani theorem represents each positive linear functional on continuous functions of compact support by such a measure.
Lebesgue measure, Haar measures and the usual measures of probability on Polish spaces are Radon measures.

## Explanations

### L1 human

A Radon measure is a measure that gives finite mass to small regions and can be read off from compact sets.
Length on the real line is the basic example.
Such a measure is determined by how it integrates continuous functions that vanish outside a bounded region.
This links measure theory with functional analysis.