Sequence of functions uniformly integrable but not bounded Find a sequence $\{ f_n : E \to \mathbb{R}, m(E) &lt; \infty\}$ of pointwise convergent functions that is uniformly integrable and not bounded by a single integrable function (i.e. there is no integrable function $g$ such that $|f_n| \leq g$ for all $n$). Show your Claim. Show that, for each example that one can find in the above problem, the limit function is integrable.

I was thinking $f_n =\frac{1}{n} \chi_{[0,n]}$ but I am not sure, anyone can help? Thank.
Check $E = (0, 1]$, $f_n(x) = \frac{n}{\log n} I_{(0, n^{-1})}(x), n = 1, 2, \ldots$. And $m$ is the Lebesgue measure on $E$.
You are on the right track, but an even simpler, more degenerate case can happen, an example that is a nice counterexample to a lot of things in analysis: set $f_n=\chi_{[n,n+1]}$. Then $\{f_n\}$ is obviously uniformly integrable, converges pointwise a.e. to $0$, and the "best" function that dominates is $g\equiv1$, which is of course not integrable.