= Solution
The <monotone convergence theorem> states that, for nonnegative measurable $f_n$ on a <Lebesgue measurable set> $E$ with $f_n\uparrow f$ <almost everywhere>,
$$
\int_Ef_n\uparrow\int_Ef,
$$
allowing infinity. <Fatou's lemma> states that, for nonnegative measurable $f_n$,
$$
\int_E\liminf_nf_n\leq\liminf_n\int_Ef_n.
$$
Now write $C=\sup_n\int_E|g_n|<\infty$. A pointwise <limit superior> of measurable functions is measurable, and it agrees with $g$ off the exceptional <null set>. By <completeness of Lebesgue measure>, assigning any values on that null set preserves measurability. Therefore $g$ is measurable. Since $|g_n|\to|g|$ almost everywhere, Fatou's lemma gives
$$
\boxed{\int_E|g|\leq\liminf_n\int_E|g_n|\leq C<\infty.}
$$
Thus $g$ is finite almost everywhere and is a <Lebesgue integrable function>. This is an <almost-everywhere limit under a uniform L1 bound>. The argument proves integrability of the limit, not convergence in <L1 space>.
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