---
tag: DYR1QFNGNJ
kind: concept
name: Borel measure
msc: [28C15, 28A12]
specializes: [PR11BZQQPG]
sources:
  - url: https://en.wikipedia.org/wiki/Borel_measure
same_as:
  wikidata: Q892585
  wikipedia: Borel measure
---

## Summary

A Borel measure on a topological space is a [[measure|PR11BZQQPG]] defined on the [[Borel sets|0WQXY32Q8D]], the sets generated by the open sets.
It is the natural kind of measure on spaces such as $\R^n$, manifolds and topological groups, where open sets and continuous functions are given.
Every continuous real function is measurable for a Borel measure.
Some authors also require a Borel measure to be finite on compact sets.

## Explanations

### L1 human

On a space with a notion of open sets, the Borel sets are all the sets one can build from open sets by complements and countable unions.
A Borel measure assigns a size to each of these sets.
Length on the real line, area in the plane and the point mass at a point are Borel measures.
They give sizes to intervals, open discs, closed sets, countable sets and much more.