This is the Excercise 1.12 of Rudin's Real and Complex Analysis:
Suppose $f\in L^1(\mu)$. Prove that to each $\epsilon>0$ there exists a $\delta>0$ such that $\int_{E}|f|d\mu<\epsilon$ whenever $\mu(E)<\delta$.
This problem likes the uniformly continuous property. I tried to prove it by making contradiction, but I can't figure it out. Thanks!