Finite Borel Measures on a Locally Compact Separable Metric Space Are Radon

In the setting of Radon Measures on Locally Compact Metric Spaces: Standing Notation, let $(X,d)$ be a locally compact metric space that is separable.

(i) Inner regularity by compact sets. Let $\mu$ be a finite Borel measure on $X$. For every $B\in\mathcal{B}(X)$,

$$\mu(B)=\sup\{\mu(K):K\subseteq X\text{ compact},\ K\subseteq B\}.$$

(ii) Radon. Every finite Borel measure $\mu$ on $X$ is a Radon measure.

(iii) Signed Radon. Every finite signed Borel measure $\nu$ on $X$ is a finite signed Radon measure.