Proof of Continuous Mapping Theorem for Weak Convergence of Finite Borel Measures

Throughout, $h$ is $\mathcal{B}(S)/\mathcal{B}(T)$-measurable (Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iv)), so that $\mu_{n}\circ h^{-1}$ and $\mu\circ h^{-1}$ are finite Borel measures on $T$ (Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (i)), and functions in $C_{b}(S)$ or $C_{b}(T)$ are Borel measurable and integrable against every finite Borel measure (Bounded Continuous Functions on a Metric Space: Measurability, Integrals, Operations, Lipschitz Functions and Approximation of Closed Sets (i), Bounded Continuous Functions on a Metric Space: Measurability, Integrals, Operations, Lipschitz Functions and Approximation of Closed Sets (ii)).

Step 1 (change of variables). Let $f\in C_{b}(T)$. Then $f\circ h\in C_{b}(S)$ by Bounded Continuous Functions on a Metric Space: Measurability, Integrals, Operations, Lipschitz Functions and Approximation of Closed Sets (v), and the change-of-variables formula, Image Measures and Measures with a Density: Measures, Integrals and Change of Variables (iii), gives, for $\nu$ any of $\mu,\mu_{1},\mu_{2},\dots$,

$$\int_{T}f\,d(\nu\circ h^{-1})=\int_{S}f\circ h\,d\nu .$$

Step 2 (conclusion). Since $f\circ h\in C_{b}(S)$ and $\mu_{n}\Rightarrow\mu$, Weak Convergence of Finite Borel Measures on a Metric Space (i) gives $\int_{S}f\circ h\,d\mu_{n}\to\int_{S}f\circ h\,d\mu$; by Step 1 this is $\int_{T}f\,d(\mu_{n}\circ h^{-1})\to\int_{T}f\,d(\mu\circ h^{-1})$. As $f\in C_{b}(T)$ was arbitrary, $\mu_{n}\circ h^{-1}\Rightarrow\mu\circ h^{-1}$ by Weak Convergence of Finite Borel Measures on a Metric Space (i) ($(\mu_{n}\circ h^{-1})$ is a sequence in the finite Borel measures on $T$, which form a set by The Measures, the Finite Measures and the Probability Measures on a Measurable Space Form Sets (ii)).