Let $(E,\mathcal E,\mu)$ be a finite [measure space](/page/Measure%20Space), and let $\mathcal F\subset L^1(E,\mathcal E,\mu)$. Suppose there exist a constant $\varepsilon_0>0$, a sequence $(f_n)_{n\in\mathbb N}$ in $\mathcal F$, and a sequence $(A_n)_{n\in\mathbb N}$ in $\mathcal E$ such that