Let $\mu$ be a finite measure on a measurable space $, and let $ be a bounded sequence in ^1(X, \mu)$. The following are equivalent:\n\begin{align*}\n&\text{(i) } (f_k) \text{ is uniformly integrable;} \\\n&\text{(ii) } (f_k) \text{ is relatively weakly sequentially compact in } L^1(X, \mu),\n\end{align*}\nmeaning every subsequence of $ has a further subsequence that converges weakly in ^1(X, \mu)$.