= Solution
Apply <monotone convergence> to $A_n=\sum_{k=1}^n|a_k|$. Its limit $A=\sum_{k\ge1}|a_k|$ has integral $\sum_k\int|a_k|<\infty$, so $A<\infty$ <almost everywhere>. Thus the original series converges absolutely <almost everywhere>. Its partial sums are <bounded> in absolute value by the <integrable function> $A$, and their limit $s$ satisfies $|s|\le A$. <Dominated convergence> gives
$$
\boxed{\int s=\lim_n\sum_{k=1}^n\int a_k=\sum_{k=1}^{\infty}\int a_k.}
$$
Assigning any value on the null exceptional set defines an <integrable> representative of $s$.
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