Support of a Borel Measure on a Metric Space

Let $(X,d)$ be a metric space and let $\mathcal{B}(X,d)$ be the Borel $\sigma$-algebra of $(X,d)$, which contains every subset of $X$ that is open in $(X,d)$. Let $\mathbb{R}$ be the real numbers, with the order $\le$ and the associated strict order $<$ of their ordered field structure.

Let $\mu$ be a Borel measure on $(X,d)$; its values lie in $[0,\infty]$, with the order of $[0,\infty]$ fixed as in Measure, Measure Space, and Probability Measure. For $x\in X$ and $r\in\mathbb{R}$ with $0<r$ the open ball $B_{d}(x,r)$ is open in $(X,d)$ by Open Ball in a Metric Space is Open and hence belongs to $\mathcal{B}(X,d)$, so that $\mu(B_{d}(x,r))$ is defined.

(Support)(i) The support of $\mu$ is the subset

$$\operatorname{supp}\mu=\bigl\{x\in X:\ 0<\mu\bigl(B_{d}(x,r)\bigr)\ \text{ for every }r\in\mathbb{R}\text{ with }0<r\bigr\}$$

of $X$.