Continuity of integrals in the product of weak topologies

Question

Consider two compact metrizable spaces $X$ and $Y$ and the sets of probability measures $\mathcal{M}(X)$ and $\mathcal{M}(Y)$ equipped with the weak topology. Furthermore let $f:X\times Y\to\mathbb{R}$ be continuous. Is the integral product map $(\mu,\nu)\mapsto\int f(x,y)\mu(\mathrm{d}x)\nu(\mathrm{d}y)$ as a map $\mathcal{M}(X)\times\mathcal{M}(Y)\to\mathbb{R}$ continuous wrt. the product topology of the weak topologies? I think that should be the case but I do not have an idea how to show it.

Answer 1

Let $\mu_n\to \mu$ and $\nu_n\to \nu$. Note that, by the Skorokhod representation, this is equivalent to having a pair of probability spaces $(\Omega_1,\mathcal{F}_1,P_1)$ and $(\Omega_2,\mathcal{F}_2,P_2)$ and random variables $Z_n,Z:\Omega_1\to X$ and $W_n,W:\Omega_2\to X$ such that $Z_n\to Z$ $P_1$-a.s. and $W_n\to W$ $P_2$-a.s.

Accordingly $(Z_n,W_n)\to (Z,W)$ $P_1\otimes P_2$-a.s. and we conclude that $\mu_n\otimes \nu_n\to \mu \otimes \nu$ weakly in $\mathcal{M}(X\times Y)$. That, however, exactly means that $$ \int f(x,y) \mu_n(dx)\nu_n(dy)=\int f(x,y) \mu_n\otimes \nu_n(dx\times dy)\to\int f(x,y) \mu\otimes \nu(dx\times dy), $$ which is what you wanted to show (since the weak topology is metrisable).