Weak Convergence of Finite Borel Measures on a Metric Space

Let $(X,d)$ be a metric space, let $(\mu_n)_{n\in\mathbb{N}}$ be a sequence of Borel measures on $(X,d)$ with $\mu_n(X)<\infty$ for every $n$, and let $\mu$ be a Borel measure on $(X,d)$ with $\mu(X)<\infty$. Regard $\mathbb{R}$ as a metric space with the absolute-value metric.

We say that $(\mu_n)_{n\in\mathbb{N}}$ converges weakly to $\mu$, written $\mu_n\Rightarrow\mu$, if for every bounded function $f:X\to\mathbb{R}$ that is continuous on $X$, the sequence of integrals $\bigl(\int_X f\,d\mu_n\bigr)_{n\in\mathbb{N}}$ converges to $\int_X f\,d\mu$.