---
tag: HQFZQ0F991
kind: concept
name: Regular measure
msc: [28C15]
specializes: [28E5Q5BM69, 33C1KEK3SC]
sources:
  - url: https://en.wikipedia.org/wiki/Regular_measure
same_as:
  wikidata: Q1428110
  wikipedia: Regular measure
---

## Summary

A regular measure on a topological space is a Borel measure that is both [[inner regular|28E5Q5BM69]] and [[outer regular|33C1KEK3SC]].
Each measurable set can then be approximated from inside by compact sets and from outside by open sets.
Regularity ties the measure to the topology and allows integrable functions to be approximated by continuous functions.
Conventions differ between authors: some approximate from inside by closed sets rather than compact sets.

## Explanations

### L1 human

For a regular measure, every set of finite measure can be squeezed between a compact set inside it and an open set around it.
The measure between the two can be made as small as one likes.
Length on the real line is the main example.
For the interval $[0, 1)$, the compact sets $[0, 1 - \varepsilon]$ inside and the open sets $(-\varepsilon, 1)$ outside have lengths as close to $1$ as one likes.