Could you give me an example about:
"a collection of functions that is uniformly integrable but each (or some) function in the collection is not integrable."
This sounds counterintuitive? However according to Royden's "Real analysis":
Definition of "uniform integrability": A family $\mathcal F$ of measurable functions on $E$ is said to be uniformly integrable over $E$ if for every $\epsilon >0$ there is a $\delta >0$ such that for each $f \in \mathcal F$, if $A\subset E$ is measurable and $m(A)<\delta$, then $\int_A|f|<\epsilon$.
Property of integrability: When $m(E)<\infty$, $f$ is (Lebesgue) integrable over $E$ if and only if for every $\epsilon >0$ there is a $\delta >0$ such that if $A\subset E$ is measurable and $m(A)<\delta$, then $\int_A|f|<\epsilon$
When $m(E)=\infty$ then there is only one direction $\Rightarrow$ in the proposition about integrability right above. So I think my desired counter example should be about a collection of functions uniformly integrable on a set of infinite measure.