Support of a measure

Remarks

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

Example: let $\delta_0$ be the Dirac measure at $0$ on $\R$. Every neighbourhood 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 $\mu(X \setminus \operatorname{supp} \mu) = 0$.