Closed–Open Regularity of a Finite Borel Measure on a Metric Space

In the setting of Radon Measures on Locally Compact Metric Spaces: Standing Notation, let $(X,d)$ be a metric space and let $\mu$ be a finite Borel measure on $X$.

(i) Closed–open approximation. For every $B\in\mathcal{B}(X)$ and every real number $\varepsilon>0$ there are a closed set $F\subseteq X$ and an open set $V\subseteq X$ with $F\subseteq B\subseteq V$ and

$$\mu(V\setminus F)<\varepsilon .$$