Let $(E,\mathcal E,\mu)$ be a [measure space](/page/Measure%20Space), let $\mathcal F,\mathcal G\subset L^1(E,\mathcal E,\mu)$ be uniformly integrable families, and let $a,b\in\mathbb R$. Suprema over empty indexing families in the defining estimates for [uniform integrability](/page/Uniform%20Integrability) are interpreted as $0$. Define

\begin{align*} a\mathcal F+b\mathcal G=\{af+bg:f\in\mathcal F,\ g\in\mathcal G\}\subset L^1(E,\mathcal E,\mu). \end{align*}

Then $a\mathcal F+b\mathcal G$ is uniformly integrable.

Moreover, if

\begin{align*} \mathcal H=\{h\in L^1(E,\mathcal E,\mu):\text{there exists }f\in\mathcal F\text{ such that } |h|\le |f|\ \mu\text{-a.e.}\}, \end{align*}

then $\mathcal H$ is uniformly integrable.