How to understand uniform integrability?

Question

From the definition to uniform integrability, I could not understand why "uniform" is used as qualifier. Can someone please enlighten me?

Answer 1

Many arguments in probability use truncation: take a random variable $X$ and define $X^k := X1_{|X| \leq k}$. This allows us to handle "most of $X$" on a compact set $K:= [-k,k]$ if $X$ is integrable, as $E|X-X^k| < \epsilon$ for sufficiently large $k$. In other words, $$\lim_{k \rightarrow\infty} E|X1_{|X| > k}| = 0.$$ The point of uniform integrability is to have a single $k$ which works for a family $\{X_\alpha\}_{\alpha \in A}$ of random variables. That is, for any $\epsilon > 0$, there is a single, uniform $k$ for which $E|X_\alpha 1_{|X| > k}| < \epsilon$. Hence, $$\lim_{k\rightarrow\infty} \sup_{\alpha \in A} E|X1_{|X| > k}| = 0.$$