Suppose I have a family $F:=\{f_\alpha\}$, $\alpha \in J$ (index set) of positive functions, a function $L$ increasing, with values in $\mathbb{R}$ such that $L^+(F):=\{L^+(f_\alpha);\alpha \in J\}$ is uniformly integrable. Where the $^+$ denotes the positive part of $L$. Can I use this to apply Dominated convergence to interchange a limit and Expectation of the form:
$\lim E[L(f_n)]\overset{!}{=}E[\lim L(f_n)]$
? Can this be done? And if so, why exactly?
hulik