Vague Convergence of Finite Signed Borel Measures

In the setting of Weak-Star Compactness and Vague Convergence: Standing Notation, let $(X,d)$ be a locally compact metric space, let $(\nu_{m})_{m\in\mathbb{N}}$ be a sequence in $M(X)$, and let $\nu\in M(X)$.

(i) Vague convergence. $(\nu_{m})$ converges vaguely to $\nu$ if the real sequences of integrals converge as follows:

$$\int_{X}f\,d\nu_{m}\to\int_{X}f\,d\nu\qquad\text{for every }f\in C_{c}(X)$$

(each $f\in C_{c}(X)$ is Borel measurable with $|f|\le\|f\|_{u}$, so it is integrable with respect to each $\nu_{m}$ and to $\nu$ by Properties of the Integral with Respect to a Finite Signed Measure: Linearity, Bounds, Bounded Functions, Linearity in the Measure, Indicators and Finite Measures (iii)).