What is the largest class of functions $f$ such that the map $\pi \mapsto \int_{\mathbb{R}^n} f \ d\pi$ is continuous, where $\pi$ is a probability measure? Further, what if $\pi$ is a probability measure with finite first moments in each marginal? That is, $\pi$ has marginals $\mu_i$ with finite first moments on each $\mathbb{R}_i$, $\mathbb{R}^n = \mathbb{R}_1 \times ... \times \mathbb{R}_n$. I am taking the weak topology.
I know that the first class contains continuous bounded functions, and the second contains functions $f(x_1, ..., x_n)$ bounded by $K(1 + \lvert x_1 \rvert + ... + \lvert x_n \rvert$), but I want to know if I can expand these.
Thanks!