= Summable L1 norms imply almost-everywhere absolute convergence
{title2=$\sum_n\|a_n\|_1<\infty\Longrightarrow\sum_n|a_n(x)|<\infty\text{ a.e.}$}
Suppose measurable $a_n$ on a <Lebesgue measurable set> $E$ satisfy $\sum_n\int_E|a_n|<\infty$. The nonnegative partial sums $S_N=\sum_{n\leq N}|a_n|$ increase to a measurable extended-valued $S$. The <monotone convergence theorem> gives $\int_ES=\sum_n\int_E|a_n|<\infty$, forcing $S<\infty$ <almost everywhere>. Each $a_n$ is finite almost everywhere too; discard the union of these countably many <null sets> when defining signed partial sums. The <series> $\sum_na_n$ then converges absolutely outside a null set.
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