= Solution
Let $S_N=\sum_{k=0}^N a_k$. These are nonnegative measurable functions increasing to the extended-valued sum $S$. <Monotone convergence> gives
$$
\boxed{\int S=\sum_{k=0}^\infty\int a_k<\infty.}
$$
A <nonnegative function> with finite integral cannot be infinite on a set of positive measure. Hence $S(x)<\infty$ <almost everywhere>, which is exactly almost-everywhere convergence of the nonnegative series. One may redefine the sum arbitrarily on the exceptional <null set> to obtain an ordinary <integrable> representative.
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