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I have seen these two definitions for uniform integrability, and I want to show that they are equivalent.

Definition 1

A collection of random variables $X_i, i \in I$ is uniformly integrable if for every $\epsilon>0$ there exists a $K$ such that $$E[|X_i|\cdot1_{|X_i|\ge K}]\le \epsilon,$$ for all $i \in I$.

Definition 2

A collection of random variables $X_i, i \in I$ is uniformly integrable, if there exists an $M$ such that

$$E[|X_i|]\le M,$$ for all $i \in I.$ And if for every $\epsilon>0$, there exists a $\delta$ so that if $P(A)<\delta$, then $$E[|X_i|\cdot 1_A]\le \epsilon,$$ for all $i \in I.$

Do you see how to prove this? I am able to show that definition 1 gives us a bound $M$ on the expectation, but the rest of the implication I am not able to show. I am also not able to go from definition 2 to definition 1.

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Page 214 in the reference in here https://www.usb.ac.ir/FileStaff/5678_2018-9-18-12-55-51.pdf

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