---
tag: ZEH0N3N467
kind: definition
name: Support of a measure
msc: [28C15, 28A12]
defines: FQTXV6BQZG
uses: [DYR1QFNGNJ, 0CQHFPGJ45, 0ZCY0Z80B6, 02BH3MCBQB, 0ACRGYNHSR]
sources:
  - url: https://en.wikipedia.org/wiki/Support_(measure_theory)
---

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]] and let $\mu$ be a [[Borel measure|DYR1QFNGNJ]] on $X$.
The *support* of $\mu$ is the set
$$\operatorname{supp} \mu = \set{x \in X : \mu(N) > 0 \text{ for every open neighbourhood } N \text{ of } x}.$$

## Remarks

Equivalently, the complement of $\operatorname{supp} \mu$ is the union of all [[open sets|02BH3MCBQB]] $U$ with $\mu(U) = 0$.
So the support is always a closed set.

Example: let $\delta_0$ be the [[Dirac measure|0ACRGYNHSR]] at $0$ on $\R$.
Every [[neighbourhood|0ZCY0Z80B6]] of $0$ contains $0$, so it has measure $1$, and $0 \in \operatorname{supp} \delta_0$.

Non-example: the point $x = 1$ is not in $\operatorname{supp} \delta_0$.
The open neighbourhood $(1/2, 3/2)$ of $1$ does not contain $0$, so its measure is $0$.
Thus $\operatorname{supp} \delta_0 = \set{0}$.

For Lebesgue measure on $\R^n$, the support is all of $\R^n$.
If $X$ is second countable, or if $\mu$ is a Radon measure, then $\mu(X \setminus \operatorname{supp} \mu) = 0$.

## Explanations

### L1 human

The support collects the points where the measure has mass in every small region around them.
Removing the open regions of zero mass leaves exactly the support.