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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

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    What is E(L(f)) if f is a function and L a functional with real values?2012-11-24

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If the question is whether $\lim \mathbb E(X_n)=\mathbb E(X)$ for every random variables $X_n$ and $X$ such that $(X_n^+)_n$ is uniformly integrable and $X_n\to X$ almost surely, then the answer is obviously not.

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    And you called the $f_\alpha$'s *functions* to indicate that they were in fact *random variables*? Anyway, the kerfuffle with $L$ and the $f_\alpha$'s only confuses things, in the end there is a sequence $L(f_n)$ of random variables and a hypothesis about $L(f_n)^+$ hence my post corresponds to the exact setting you have in mind. Re your comment/new question about $L$ being increasing, one can always assume that $L$ is the identity hence this point is moot. All in all, you seem to want to have your cake and eat it, that is, to omit the hypothesis of DCT and keep its conclusion. This won't do.2012-11-25