Here is another solution.
Define $E_j=\{x:f(x)\geq j \},\forall j\in \mathbb N$ .
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_{E_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 another exercise of Rudin to get $\lim_{j\rightarrow0}\int_{A_j}|f| d\mu=0 $.