---
tag: FQTXV6BQZG
kind: concept
name: Support of a measure
msc: [28C15, 28A12]
sources:
  - url: https://en.wikipedia.org/wiki/Support_(measure_theory)
same_as:
  wikidata: Q2709287
  wikipedia: Support (measure theory)
---

## Summary

The support of a [[measure|PR11BZQQPG]] on a [[topological space|0CQHFPGJ45]] is the set of points all of whose open neighbourhoods have positive measure.
It is a closed set, and it tells where the measure really lives.
For a [[Radon measure|HZNEQQBCW4]] the complement of the support has measure zero.

## Explanations

### L1 human

Think of a measure as mass spread over a space.
The support is the place where some mass is found near every point, however closely one looks.
Points far from any mass are outside the support.
For example, the unit point mass at $0$ on the real line has support $\set{0}$.
The length measure on the real line has support the whole line, since every open interval has positive length.