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- Jun 20, 2014

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Let $(X_n)$ be a sequence of $L^1$ random variables on a probability space $(\Omega, \Bbb P)$. Let $f$ be a continuous, nondecreasing function from $[0,\infty)$ onto itself such that

1. $\Bbb E[f(|X_n|)]$ is uniformly bounded

2. $\dfrac{f(x)}{x}\to \infty$ as $x\to \infty$

Show that $(X_n)$ is uniformly integrable.

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