= Solution
The partial sums $s_n=\sum_{k=1}^n a_k$ are nonnegative <measurable function>[measurable functions] and increase to an extended-valued function $s$. By the <monotone convergence theorem>,
$$
\int_{\mathbb R}s(x)\,dx
=\lim_{n\to\infty}\sum_{k=1}^n\int_{\mathbb R}a_k(x)\,dx
=\sum_{k=1}^\infty\int_{\mathbb R}a_k(x)\,dx<\infty.
$$
A nonnegative function with finite <Lebesgue integral> cannot equal $+\infty$ on a set of positive measure: otherwise its <integral> would exceed $M$ times that measure for every $M>0$. Thus the <series> converges to a finite value almost everywhere. Redefine it to be zero on the exceptional <null set> if a finite-valued representative is desired. The resulting function is <Lebesgue integrability>[Lebesgue integrable], with
$$
\boxed{\int\sum_{k=1}^\infty a_k=\sum_{k=1}^\infty\int a_k.}
$$
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