I want to understand uniform integrability in terms of Lebesgue integration According to my Real Analysis textbook, a family $\scr{F}$ of measurable functions on $E$ is said to be uniformly integrable over $E$ provided for each $\epsilon$ $>$ $0$, there is a $\delta$ $>$ $0$ such that for each $f$ $\in$ $\scr{F}$, if $A$ $\subseteq$ $E$ is measurable and $m(A)$ $<$ $\delta$, then $\int_{A}$$|f|$ $<$ $\epsilon$. That is fine, but my textbook doesn't really give any good examples or counterexamples. I am studying for a final exam in Real Analysis, and would like some input in regards to any examples or counterexamples for uniform integrability of Lebesgue integrable functions. Thanks!
 A: You might be able to find more examples of UI and non-UI functions from probability books. But the definition there is different, and not equivalent to yours (similar ideas though).
Here are some examples that you can find/adapt from Wikipedia. You still need to prove them because the definition there is different. 


*

*Any finite set of Lebesgue integrable functions is U.I. (because of absolute continuity of
integral.) 

*Any family bounded by an $L^1$ function: $\{f\in L^1: |f|\le g\}$ where $g\in L^1$ is U.I.

*$\{f\in L^p:\|f\|_{L^p}\le C\}$ with $p>1$ is U.I. (because of Holder's inequality)


Examples of non-U.I. families should not be too hard. You can construct a sequence of functions with the "mass" more and more concentrated. For example, consider the family of functions $\{f_n(x)=n \mathbf{1}_{[0,\frac{1}{n}]}:n=1,2,\cdots\}$. It's easy to use your definition to prove that this family is not U.I. (But it is U.I. in the sense defined in probability theory, which you can find from Wikipedia.) 
