Let $(E,\mathcal E,\mu)$ be a [measure space](/page/Measure%20Space). Let $(f_n)_{n\in\mathbb N}$ be a sequence in $L^1(E,\mathcal E,\mu)$, and let $f\in L^1(E,\mathcal E,\mu)$. Assume that
Then $\mathcal F$ is uniformly integrable in the small-set sense: for every $\varepsilon>0$ there exists $\delta>0$ such that, for every $A\in\mathcal E$ with $\mu(A)<\delta$,