Generalization of the dominated convergence theorem There exist theorems, such as dominated convergence theorem, monotone convergence theorem, for a sequence of integrals in which both integrand and measure change in $n$?
In other words, there is a theory about the convergence of sequences of integrals like
$\int f_n d\mu_n\xrightarrow{n\to+\infty}$
If yes, could someone give me some reference?
 A: Sure, here are two examples:
Let $X$ be a separable metrizable space endowed with the Borel $\sigma$-algebra, 
$\langle f_n\rangle$ a sequence of bounded continuous functions on $X$ that converges uniformly to $f$, and let $\langle\mu_n\rangle$ be a sequence of Borel probability measure such that for some Borel probability measure $\mu$ and each bounded continuous function $g$ on $X$, $\lim_n\int g~\mathrm d\mu_n=\int g~\mathrm d\mu$. Then $\lim_n\int f_n~\mathrm d\mu_n=\int f~\mathrm d\mu$. This notion of convergence of measures is weak or narrow convergence.
Let $(X,\mathcal{X})$ be a measurable space, and let $\langle \mu_n\rangle$ be a sequence of probability measures converging in variation to $\mu$, that is for each $\epsilon>0$, we have
$$\sup_{A\in\mathcal{X}}|\mu_n(A)-\mu(A)|<\epsilon$$
for $n$ large enough.  Let $\langle f_n\rangle$ be a uniformly bounded sequence of measurable functions converging pointwise to $f$. Then $\lim_n\int f_n~\mathrm d\mu_n=\int f~\mathrm d\mu$. The proof is an exercise in epsilontics and approximating measurable functions by simple functions. 
