Let \(\left(f_{i}ight)_{i \in I}\) be a family of uniformly integrable functions and let \(\left(u_{i}ight)_{i \in I} \subset \mathcal{L}^{1}(\mu)\) be some further family such that \(\left|u_{i}ight| \leqslant\left|f_{i}ight|\) for every \(i \in I\). Then \(\left(u_{i}ight)_{i \in I}\) is uniformly integrable. In particular, every family of functions \(\left(u_{i}ight)_{i \in I}\) with \(\left|u_{i}ight| \leqslant g\) for some \(g \in \mathcal{L}_{+}^{1}(\mu)\) is uniformly integrable.