Helly Selection Theorem for Finite Signed Borel Measures on a Locally Compact Separable Metric Space

In the setting of Weak-Star Compactness and Vague Convergence: Standing Notation, let $(X,d)$ be a locally compact metric space that is separable, let $C\in\mathbb{R}$ with $C\ge0$, and let $(\nu_{m})_{m\in\mathbb{N}}$ be a sequence in $M(X)$ with $\|\nu_{m}\|\le C$ for every $m\in\mathbb{N}$.

(i) Selection. There are a subsequence $(\nu_{m_{j}})_{j\in\mathbb{N}}$ of $(\nu_{m})$ and a $\nu\in M(X)$ with $\|\nu\|\le C$ such that $(\nu_{m_{j}})$ converges vaguely to $\nu$.

(ii) Nonnegative measures. If moreover $\nu_{m}(A)\ge0$ for every $m\in\mathbb{N}$ and every $A\in\mathcal{B}(X)$, then every $\nu\in M(X)$ to which some subsequence of $(\nu_{m})$ converges vaguely satisfies $\nu(A)\ge0$ for every $A\in\mathcal{B}(X)$.