Limit theorems in measure theory

From probability theory/measure theory we know set of theorems such as Monotone convergence, dominated convergence or conditions like uniform integrability which deals with the general question of interchanging limits and integration. In these cases we have either measurable functions $f_n$ converging to $f$ or measures $\pi_k$ converging to $\pi$ and we see under what condition for instance $\int f_n\rm d \pi \to \int f\rm d \pi$ or $\int f\rm d \pi_k\to \int f\rm d \pi$.

My question is what happens if those two are mixed. Namely if we have measurable functions $f_n$ converging to $f$ and measures $\pi_k$ converging to $\pi$ then what we can say about limits of $\int f_n\rm d \pi_n$. A particularly interesting case is when $f_n$ is implicitly a functional of $\pi_n$, which means that by changing the measure $f_n$ will change too.

I have looked into many classical books on measure theory and probability theory (Rudin, Billingsley, Feller, Durrett, Halmos, etc), but could not find an answer to this.

Any help is appreciated.

Solution 1:

A sufficient condition is that $\pi_n$ are bounded and converge to $\pi$ in total variation and that $f_n$ are continuous and uniformly bounded ($\sup_n \|f_n\|_\infty < \infty$). Note that Scheffé's Lemma automatically gives convergence in total variation if all $\pi_n$ are absolutely continuous (with respect to the Lebesgue measure, or any other one) and the mass of $\pi_n$ converges to the one of $\pi$ (for instance they all are probability measures).

In a given setting, maybe it is possible to directly study whether the convergence $\lim_{n\to\infty} \int f_n d\pi_k =\int f d\pi_k$ is uniform with respect to $k$.