= Solution
Let $S_N=\sum_{n=1}^N|a_n|$. These are nonnegative measurable functions with $S_N\uparrow S=\sum_{n\geq1}|a_n|$. The <monotone convergence theorem> and linearity of the finite sum give
$$
\int_ES=\lim_{N\to\infty}\int_ES_N
=\sum_{n\geq1}\int_E|a_n|<\infty.
$$
Hence $S<\infty$ <almost everywhere>: if $S=\infty$ on a set of positive measure, the integral would exceed every finite constant. Also each $a_n$ is finite almost everywhere, since its absolute integral is finite. Outside the union of these countably many <null sets>, all terms are real and
$$
\boxed{\sum_{n=1}^\infty|a_n(x)|<\infty.}
$$
Thus the signed series converges absolutely almost everywhere, as asserted by <summable L1 norms imply almost-everywhere absolute convergence>. Removing the common null set avoids any undefined $+\infty-\infty$ in signed partial sums.
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