= Solution
The partial sums $S_n=\sum_{k=1}^na_k$ are nonnegative measurable functions and increase pointwise to the extended-valued sum $S=\sum_{k=1}^\infty a_k$. The <monotone convergence theorem> and finite linearity give
$$
\int S\,dx=\lim_n\sum_{k=1}^n\int a_k\,dx
=\sum_{k=1}^\infty\int a_k\,dx<\infty.
$$
A nonnegative <measurable function> with finite integral is finite <almost everywhere>; if it were infinite on a positive-measure set, its integral would be infinite. Define it arbitrarily on the null exceptional set if a real-valued representative is desired. Hence the series converges <almost everywhere> to an integrable function and
$$
\boxed{\int\sum_{k=1}^\infty a_k(x)\,dx=\sum_{k=1}^\infty\int a_k(x)\,dx.}
$$
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