= Solution
Let $S_N(x)=\sum_{n=1}^N|f_n(x)|$. These are nonnegative <measurable functions> increasing to the extended-valued <function> $S(x)=\sum_{n=1}^{\infty}|f_n(x)|$. The <monotone convergence theorem> gives
$$
\int_ES(x)\,dx=\lim_{N\to\infty}\sum_{n=1}^N\int_E|f_n(x)|\,dx
=\sum_{n=1}^{\infty}\|f_n\|_1<\infty.
$$
Therefore $S$ is finite <almost everywhere>: if $S=\infty$ on a <measurable set> of positive <Lebesgue measure>, its <integral> would exceed every finite bound. Hence the <series> $\sum_nf_n(x)$ converges absolutely outside the <null set> $N=\{S=\infty\}$. This is <summable L1 norms imply almost-everywhere absolute convergence>.
Define $s(x)=\sum_nf_n(x)$ off $N$ and set $s=0$ on $N$. The <function> is <Lebesgue measurable>, being the pointwise <limit> of measurable partial sums off a measurable <null set>. Since $|s|\leq S$ <almost everywhere>,
$$
\|s\|_1\leq\int_ES<\infty,
$$
so \b[the sum is <Lebesgue integrable>].
For $s_N=\sum_{n=1}^Nf_n$, the absolute tail satisfies
$$
|s-s_N|\leq\sum_{n=N+1}^{\infty}|f_n|\qquad\text{almost everywhere}.
$$
Apply the <monotone convergence theorem> to the nonnegative tail partial sums. We obtain
$$
\|s-s_N\|_1\leq\sum_{n=N+1}^{\infty}\|f_n\|_1\longrightarrow0.
$$
Thus the partial sums converge in <L1 norm>. The <triangle inequality for an integral> yields
$$
\left|\int_Es-\sum_{n=1}^N\int_Ef_n\right|\leq\|s-s_N\|_1\longrightarrow0.
$$
Consequently
$$
\boxed{\int_E\sum_{n=1}^{\infty}f_n(x)\,dx=\sum_{n=1}^{\infty}\int_Ef_n(x)\,dx.}
$$
The scalar <series> on the right also converges absolutely, since $\sum_n|\int_Ef_n|\leq\sum_n\|f_n\|_1<\infty$. This proves <integration of a series with summable L1 norms> without needing pointwise convergence on the exceptional <null set>.
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