prove that there exists $E$ such that $m(X/E)=0$ and $\lim_k(f_k(x))=0$ for all $x\in E$.

Question

$(X,S,\mu)$ is a measure space ${f_k}$ are S- measurable real- valued functions such that, $\sum_{k=1}^{\infty} \int|f_k|d\mu<\infty$, prove that there exists $E\in S$ such that $m(X/E)=0$ and $\lim_k(f_k(x))=0$ for all $x\in E$.

My work-

since $\sum_{k=1}^{\infty} \int|f_k|d\mu<\infty$

$\lim_k(\int |f_k(x)|)=0$

then, by the limit definition, for all $\epsilon>0$ there exists $K$ such that for all $k>K$

$\int |f_k(x)|<\epsilon$. I'm stuck in here.

How do I proceed from here to show that
$\lim_k(|f_k(x)|)=0$

Is there any other approach? Thank you.

Answer 1

Hint: Use the MCT to see

$$\sum \int|f_k|\,d\mu = \int(\sum |f_k|)\,d\mu.$$

Think about what must be true of $\sum |f_k|$ for the last integral to be finite.