Find a sequence $\{ f_n : E \to \mathbb{R}, m(E) < \infty\}$ 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.
Best Answer
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 $E$.