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Let $m$ be a probability measure on the compact set $W \subset \mathbb{R}^m$, so that $m(W)=1$.

Consider $f: X \times W \rightarrow \mathbb{R}_{\geq 0}$ locally bounded, $X \subseteq \mathbb{R}^n$, such that

$\forall w \in W$ $\ x \mapsto f(x,w)$ is continuous;

$\forall x \in X$ $\ w \mapsto f(x,w)$ is measurable.

Assume that for any $x \in X$, we have $\int_W f(x,w) m(dw) < \infty$.

Say if the following property is true.

There exists $\delta>0$ such that the family of functions $\{ w \mapsto f(\xi,w) \mid \xi \in \{x\} + \delta \overline{\mathbb{B}} \} $ is Uniform Integrable.

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Why the link you give should regard a compact $W$? –  Adam Jun 15 '12 at 1:50
This is still duplication of math.stackexchange.com/questions/126087/uniform-integrability. $W$ being compact changes nothing. In particular, if $f$ is locally bounded independently of $x$, then the answer to the question is yes; otherwise, no. –  William Jun 15 '12 at 2:58
I don't see the proof of either "yes" or "no". I mean, the mapping $(x,w) \mapsto f(x,w)$ is locally bounded and for all fixed $w$ the mapping $x \mapsto f(x,w)$ is continuous. –  Adam Jun 15 '12 at 5:39
I've just edited the answer in that link slightly; it should be more clear now; in particular, look at Case ii. –  William Jun 15 '12 at 18:27
Ok, now I see. Thanks. –  Adam Jun 15 '12 at 19:29

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