How are these conditions in the definition of uniform integrability different from eachother

Question

My lecture notes define uniform inegrability as follows

A family $(X_i)_{i\in I}$ of real random variables is called uniformly integrable, if

1. $\sup_{i\in I} E(|X_i|)<\infty$

2. $\sup_{i\in I}E\left(|X_i|\cdot 1_{\{|X_i|\geq k\}}\right)\rightarrow0$ for $k\rightarrow\infty$.

I'm wondering why we need these two conditions, doesn't the first one imply the second ?

Answer 1

The first condition does not imply the second.

For instance let $I=\mathbb N$ and for $i=1,2,3,\dots$ let $P(X_i=i)=\frac1{i}$ and $P(X_i=0)=1-\frac1{i}$.

Then $\mathbb E|X_i|=\mathbb EX_i=1$ for every $i$ so that $\sup_{i\in\mathbb N}\mathbb E|X_i|=1<\infty$.

But for every $k$ we have $\mathbb E|X_i|\mathbf1_{|X_i|\geq k}=1$ for $i\geq k$ so that $\sup_{i\in\mathbb N}\mathbb E|X_i|\mathbf1_{|X_i|\geq k}=1$ for every $k$.