Our textbooks says "We infer from Proposition 23 of the preceding chapter that if $F$ is a family of functions on E that is uniformly integrable and tight over E, then each function in F is integrable over E."
Let f be a measurable funciton on E. If f is integrable over E, then for each $\epsilon < 0$, there is a $\delta < 0$ for which
if $A \subset E$ is measurable and $m(A)<\delta$, then $\int_A |f| < \epsilon$.
Conversely, in the case $m(E)<\infty$, if for each $\epsilon < 0$, there is a $\delta < 0$ for which (26) [the above] holds, then f is integrable over E.
What I don't understand is...if there is a family of functions that is uniformly integrable, then how is it possible for one of the functions in that family to $*not*$ be integrable? Why do we need tightness to conclude that?
Thanks in advance