For which $f$ does $\int f d\mu_n\to\int f d\mu$?

Question

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 $E=C((\mu_n), \mu)$ ?

Answer 1

This set is not closed with respect to monotone limits in general: if $\mu_n = \delta_{1/n}$, $\mu = \delta_0$, $f_k = \mathbb{1}_{[1/k,1]}$, $f = \mathbb{1}_{(0,1]}$ then $\int f_k d\mu_n = 0 = \int f_k d\mu$ for $n$ large enough, but $\int f d\mu_n = 1 \ne 0 = \int f d\mu$ for all $n$.

It is closed under uniform convergence, though: if $f_k$ converge to $f$ uniformly, then the sequence $(\int f_k d\mu_n : n > 0)$ converge to the sequence $(\int f d\mu_n : n > 0)$ uniformly, and also $\int f_k d\mu$ converges to $\int f d\mu$ (as $k \to \infty$).