= Almost-everywhere limit under a uniform L1 bound
{title2=$\int|g|\leq\liminf_n\int|g_n|\leq\sup_n\|g_n\|_1$}
Let $g_n$ be <Lebesgue measurable functions> on a <Lebesgue measurable set> $E$, converging <almost everywhere> to $g$, with $\sup_n\int_E|g_n|<\infty$. The limit has a measurable representative: use $\limsup_ng_n$ off the exceptional <null set> and zero on it. The <completeness of Lebesgue measure> makes any alternative values on that null set measurable as well. <Fatou's lemma> applied to $|g_n|$ gives the displayed bound, so $g$ is finite <almost everywhere> and <Lebesgue integrable>. A uniform <L1 norm> bound alone does not imply convergence in <L1 space>; the <strict inequality in Fatou lemma> supplies a shrinking-spike example.
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