Suppose $ f_1, f_2, \cdots$ is a collection of measurable functions which satisfy $ \sup_n \int |f_n|^{1 + \gamma}\ d\mu< \infty$ for some $ \gamma > 1$ , and $ \mu$ is a finite measure. I am being asked to show that the $ \{f_n\}$ are uniformly absolutely continuous. That is, for each $ \epsilon > 0$ there exists a $ \delta > 0$ such that for all $ A$ with $ \mu(A) < \delta$ $ $ \int_{A} |f_n| < \epsilon$ $ for all $ n \geq 1$ .

I’m not sure how to use the fact that $ \sup_n \int |f_n|^{1 + \gamma}\ d\mu< \infty$ . If I assume only that $ \sup_n \int |f_n|\ d\mu< \infty$ , it seems I can prove something weaker:

Suppose $ A_1, A_2, \cdots$ is any sequence of sets for which $ \mu(A_n) \to 0$ as $ n \to \infty$ , and let $ m \geq 1$ . Since $ f_m \cdot 1_{A_n} \to 0$ a.e. and $ |f_m \cdot 1_{A_n}| \leq |f_m|$ and $ \int |f_m| < \infty$ , the dominated convergence theorem implies that $ \int_{A_n} |f_m| \to 0$ as $ n \to \infty$ .

But I’m not sure where to go from here.