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The Dunford-Pettis Theorem (see Uniform Integrability Wiki) states that:

A class of random variables $X_n \in L^1(\mu)$ is Uniformly Integrable if and only if it is relatively weakly compact.

Now $X_n \in L^1(\mu)$ means that $\sup_n \int_\mathbb{X} |X_n(x)| \mu(dx) < \infty$.

A family $\mathcal{X}$ is relatively weakly compact if for every sequence $\{X_n\}$ in $\mathcal{X}$ there exist a subsequence $\{X_{\tilde{n}}\}$ and $X_* \in \mathcal{L}^1(\mu)$ such that $$ \lim_{\tilde n} \int_A X_{\tilde n}(x) \mu(d x) = \int_A X_*(x) \mu(dx) \quad \forall A \subseteq \mathbb{X} \ s.t. \int_A \mu(dx) > 0 $$

Given $Y: \mathbb{R}^m \times \mathbb{X} \rightarrow \mathbb{R}_{\geq 0} $ continuous in the first argument, locally bounded in the second, is the family $\{ Y(n,\cdot) \}_{n \in \mathcal{N}}$, $Y(n,\cdot) \in L^1(\mu)$, being $\mathcal{N}$ is compact, "relatively weakly compact"?

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No, $X_n \in L^1(\mu)$ means: for each $n$, $\int |X_n(x)| \mu(dx) < \infty$ –  GEdgar Apr 20 '12 at 17:05
Ok, thanks for the typo on $|\cdot|$. –  Adam Apr 20 '12 at 17:13
Is there someone who can provide examples of relatively-weakly-compact families of integrable functions? –  Adam Apr 21 '12 at 15:50
Maybe you should take a look at this note from Diestel: openstarts.units.it/dspace/bitstream/10077/4777/1/… I think this may help. –  user89421 Aug 7 '13 at 17:05

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