Weak Convergence of Finite Borel Measures is Preserved by Continuous Maps between Metric Spaces

In the setting of The Real Numbers: Standing Notation and Background, let $(Y,d_{Y})$ and $(Z,d_{Z})$ be metric spaces with Borel $\sigma$-algebras $\mathcal{B}(Y)$ and $\mathcal{B}(Z)$, and let $T:Y\to Z$ be continuous on $Y$, so that $T$ is measurable with respect to $\mathcal{B}(Y)$ and $\mathcal{B}(Z)$ by claim 3 of Borel Measurability and Bounded Integration on a Metric Space. For a Borel measure $\mu$ on $(Y,d_{Y})$, $T_{\#}\mu$ denotes the image measure $B\mapsto\mu(T^{-1}(B))$ on $(Z,\mathcal{B}(Z))$ of claim 1 of that lemma, a Borel measure on $(Z,d_{Z})$ with $T_{\#}\mu(Z)=\mu(Y)$.

¶ Let $(\mu_{j})_{j\in\mathbb{N}}$ be a sequence of Borel measures on $(Y,d_{Y})$ and $\mu$ a Borel measure on $(Y,d_{Y})$, all of finite total mass, with $\mu_{j}\Rightarrow\mu$ in the sense of weak convergence. Then $T_{\#}\mu_{j}\Rightarrow T_{\#}\mu$.