Let $(E, \mathcal{E}, \mu)$ be a measure space with $\mu(E) < \infty$, and let $\mathcal{F} \subset L^1(E, \mu)$. The family $\mathcal{F}$ is uniformly integrable if and only if there exists a convex function $\Phi: [0, \infty) \to [0, \infty)$ with $\Phi(t)/t \to \infty$ as $t \to \infty$ such that $\sup_{f \in \mathcal{F}} \int_E \Phi(|f|) \, d\mu < \infty$.