Definition 1

If $\mu$ is a measure on a locally compact space $X$, one defines the support of $\mu$, denoted $\operatorname{Supp}(\mu)$, to be the closed set complementary to the largest of the open sets in $X$ on which the restriction of $\mu$ is zero.

To say that a point $x \in X$ does not belong to the support of $\mu$ means that there exists an open neighborhood $V$ of $x$ such that the restriction of $\mu$ to $V$ is zero; to say that $x$ belongs to the support of $\mu$ therefore means that for every neighborhood $V$ of $x$, there exists a function $f \in \mathscr{K}(X; \mathbf{C})$, whose support is contained in $V$, such that $\mu(f) \neq 0$.