measure theory - For every $epsilon>0$ there exists $delta>0$ such that $int_A|f(x)|mu(dx) < epsilon$ whenever $mu(A) < delta$

Hello all mathematicians!! Again, I am struggling with solving the exercises in Lebesgue Integral for preparing the quiz. At this moment, I and my friend are handling this problem, but both of us agreed this problem is a bit tricky.

The problem is following.

Let ($X,\mathcal{A},\mu$) be a measure space and suppose $\mu$ is $\sigma$-finite. Suppose $f$ is integrable. Prove that given $\epsilon$ there exist $\delta$ such that

$$\int_A |f(x)|\mu(dx) \;\;<\;\;\epsilon$$

whenever $\mu(A) < \delta$.

Could anybody give some good idea for us? Think you very much for your suggestion in advance.

Answer

Hint:

$$\int_A|f|\leqslant x\mu(A)+\int_{\{|f|\gt x\}}|f|\qquad\&\qquad\lim_{x\to\infty}\int_{\{|f|\gt x\}}|f|=0$$