For questions about families of uniformly integrable random variables. Use the tags (measure-theory) or (probablity-theory).

Given a finite measure space $(X,\mathcal F,\mu)$, where $\mu$ is a positive measure, and $\{f_i,i\in I\}$ a family of measurable functions from $X$ to the real line, we say that $\{f_i,i\in I\}$ is uniformly integrable if $$\lim_{R\to +\infty}\sup_{i\in I}\int_{\{|f_i|>R\}}|f_i|d\mu=0.$$

We can extend this definition to infinite measure space by requiring that for each $\varepsilon\gt 0$, there is $g$ nonnegative and integrable such that $$\sup_{i\in I}\int_{\{|f_i|>g\}}|f_i|d\mu\lt\varepsilon.$$

Such families have good properties, for example if $f_n\to 0$ almost everywhere and $\{f_n,n\geqslant 1\}$ is a uniformly integrable family, then $\lVert f_n-f\rVert_1\to 0$ as $n\to\infty$.