Hi this is a problem I find and I have had problem finding the example which the problem ask I'd appreciate if someone can help me with this. Thank you.
Let $(X,\mathscr A, \mu)$ a measure space and let $\{f_k\}$ $\mathscr A$-measurable functions, give condition on $\{f_k\}$ such that $\int \sum f_k d\mu = \sum_k \int f_k d\mu$ and give an example when $\sum_k f_k (x)$ is absolutely convergent for any $x$, the $f_k \in \mathscr L ^1 (X,\mu)$ and the conclusion fails.
I'll omit almost all the proof because is really easy. If $\{f_k\}$ is $[0,\infty]$-valued, the monotone convergence theorem implies that $\int \sum f_k d\mu = \sum_k \int f_k d\mu$
If $\sum_k \int |f_k| d\mu= \int \sum_k |f_k| d\mu<\infty$ the dominated convergence theorem implies the result.
Example:
Try $f_k=\bf 1_{[2^k,2^{k+1}]}-\bf 1_{[2^{k+1},2^{k+2}]}$