Over a probability space $( X, \mathcal{B}, m )$,
1) A collection $\mathcal{F} \subset L^1 (m)$ is called uniformly integrable if for all $\epsilon > 0,\ \exists M > 1$ s. t. $\int_{|f| \geq M} |f|\,dm \leq \epsilon\ \forall f \in \mathcal{F}$.
2) A collection $\mathcal{F}$ is weakly sequentially precompact if every sequence $\{ f_n \}$ in $\mathcal{F}$ has a subsequence $\{ f_{n_k} \}$ which converges weakly to some $f \in L^1 (m)$.
Show that a collection is uniformly integrable iff it is weakly sequentially precompact in $L^1$.
(I have made a few attempts but all in vain so far. Hoping that it might be of interest to others too.)
