Question (2) on Uniform Integrability

Let $m$ be a probability measure on $W \subseteq \mathbb{R}^m$, 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:

(1) $\forall x \in X$ $\ \int_W f(x,w) m(dw) < \infty$

(2) For all $x \in X$ we have the following property.

$\forall \epsilon, \delta > 0$ $\ \exists c >0$ such that

$$\sup_{\xi \in \{x\}+\delta \overline{\mathbb{B}} } m\left( \{w \in W \mid f(\xi,w) \geq c \} \right) \leq \epsilon$$

(1) Prove that (Uniform Integrability) for any fixed $x \in X$ we have:

$\forall \epsilon >0$ $\ \exists \delta, c>0$ such that

$$\sup_{\xi \in \{x\}+\delta \overline{\mathbb{B}} } \int_{\{ f(\xi,w) \geq c \}} f(\xi,w) m(dw) \leq \epsilon$$

(2) Can assuming $(x,w) \mapsto f(x,w)$ continuous help in getting the proof?

Note: this question is a variant of Question on UI.

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Why can't you simply replace $X$ in the previous equation with $\{x\}+\delta\overline{B}$? – user31373 Jun 11 '12 at 16:58
Because I want to take $\delta$ sufficiently small in order not to additionally assume that some summation (involving $\epsilon_n$s and/or $c_n$s) is finite. I mean that one integrable function ($\xi \in \{x\}$) is also uniformly integrable. Then I want a small $\delta$-ball around $\{x\}$ such that we have UI, exploiting the assumptions. – Adam Jun 11 '12 at 17:00
Then you need some sort of continuity with respect to $x$. Otherwise it could be $f(x,w)=0$ and $f(x',w)=1/w$ when $x'\ne x$ (with $W=[0,1]$). By the way, what is the nature of $X$? You are using the sum notation, suggesting it's a vector space or a subset of it. – user31373 Jun 11 '12 at 17:10
Ok, I'll be more precise in the question. Yes, I have continuity of $f$ in the first argument. – Adam Jun 11 '12 at 17:12
Define $\epsilon_n(\delta) := \sup_{\xi \in \{x\}+\delta \overline{\mathbb{B}}} m( \{ w \in W \mid f(\xi,w) \geq 2^n \} )$ for $\delta \in \mathbb{R}_{\geq 0}$. Is $\epsilon_n(\cdot)$ a continuous function? – Adam Jun 11 '12 at 19:29