= Solution
The partial sums $S_N=\sum_{k=0}^Na_k$ are measurable, nonnegative and increasing. By the <monotone convergence theorem>,
$$
\boxed{\int\sum_{k=0}^\infty a_k=\sum_{k=0}^\infty\int a_k<\infty}.
$$
A nonnegative measurable <function> with finite integral is finite almost everywhere: if it were infinite on a positive-measure set, every constant multiple of that set's indicator would be a lower bound, forcing an infinite integral. Thus the series \b[converges almost everywhere].
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