Let $\{f_k\}$ be measurable functions such that $\sum_{k=1}^\infty\int |f_k|d\mu<\infty$ Show that $\lim_{k\rightarrow 0}f_k(x)=0$ almost everywhere.

Question

Let $\{f_k\}$ be a sequence of measurable functions such that $$\sum_{k=1}^\infty\int |f_k|d\mu<\infty$$ Show that $\displaystyle\lim_{k\rightarrow 0}f_k(x)=0$ almos everywhere.

Since $\displaystyle\sum_{k=1}^\infty\int |f_k|d\mu<\infty$ I know that $\int |f_k|d\mu\rightarrow 0$, which means that the functions $f_k$ must be tending to zero almost everywhere, but I can't find a way to prove this.

Answer 1

So by Tonelli, pairing with counting measure and $\mu$, one has \begin{align*} \int\sum|f_{k}|<\infty, \end{align*} and hence \begin{align*} \sum|f_{k}(x)|<\infty,~~~~\text{a.e.}~x, \end{align*} can you finish from here?