# Equivalent condition of uniform integrability of a sequence of random variables

Here's the definition I have for a sequence of random variables to be uniformly integrable:

$(1)$ A sequence of random variables $X_1, X_2, \ldots$ is uniformly integrable (U.I.) if for every $\epsilon>0$ there exists a $K$ such that for each $n$, $\mathbb{E}[|X_n|I\{|X_n|>K\}]<\epsilon.$

I've seen that U.I. implies the following:

$(2)$ Given $\epsilon>0$, there is a $\delta>0$ such that if $\mathbb{P}(A)<\delta$, then $\mathbb{E}(|X_n|I_A)<\epsilon$.

In the book I'm reading, the author states a fact about a condition that implies U.I. and in the proof, he just shows that $(2)$ holds. Is it the case that $(1)\iff(2)$?

Also, this is supposed to be pre-measure-theoretic, so if you can answer my question without measure theory that would be ideal. Thanks in advance for your input!

• what answer are you hoping for? the short answer is that 'yes' they are the same. – Lost1 Apr 14 '13 at 18:18
• I was hoping the answer is 'yes.' The author doesn't mention that the two are equivalent, but uses the fact that they (apparently) are! – eeeeeeeeee Apr 14 '13 at 18:21
• do you want a proof? – Lost1 Apr 14 '13 at 18:22
• That would be nice! – eeeeeeeeee Apr 14 '13 at 18:23
• Can you just take the event $A=\{|X_n|>K\}$? – eeeeeeeeee Apr 14 '13 at 18:24