Let $\mu_n$ be a sequence of probability measures, and $\mu$ some other probability measure (all on the same space). Denote $C((\mu_n), \mu)$ the space of functions $f$ such that $\int f d\mu_n\to_{n\to\infty}\int f d\mu$. Can anything be said a priori about this set? Is it necessarily closed under monotone convergence, for example?

More generally, let $E$ be a subspace of functions for which all of the required integrals are defined. When does there exist a $(\mu_n)$ and $\mu$ such that $E=C((\mu_n), \mu)$ ?