Borel measures finite on balls are outer regular by open sets and inner regular by closed sets

Let \(X\) be a metric space with Borel \(\sigma\)-algebra \(\mcb\), and let \(\mu\) be a measure on \(\mcb\) with \(\mu(B)<\infty\) for every open ball \(B\) of finite radius. Then for every \(E \in \mcb\) and every \(\varepsilon>0\):

• there exists an open set \(O\subseteq X\) with \(E \subseteq O\) and \(\mu(O\setminus E) < \varepsilon\);

• there exists a closed set \(F\subseteq X\) with \(F\subseteq E\) and \(\mu(E\setminus F) < \varepsilon\).