TITLE: Uniformly Continuous Like Property of the Integration on Measure Space
QUESTION [6 upvotes]: 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!

REPLY [2 votes]: Here is another solution.
Define $A_j=\{x:|f(x)|\geq j \},\forall j\in \mathbb N$ .
The given statement is clearly true if $|f|$ is bounded. So if we can show the integrals over the defined sets tend to zero, we are done.
Now, $ \chi_{_{A_j}} $ is a collection of monotonically decreasing sequence with limit $0$. And so is true for $  |f|  \chi _{_{A_j}}  $ with $|f|\chi _{_{A_1}}\in L^1(\mu)$. So apply Dominated Convergence Theorem to get $\lim_{j \to \infty}\int_{A_j}|f| d\mu=0 $.