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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 ${\mathrm{limsup}}_n{g}_n$ 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.

On the probability space $([0,1],\mathcal{B},\lambda )$, letFor every $x\in [0,1]$, eventually $x\notin (0,1/n)$, so $f_n(x)\to 0$. Nevertheless,for every $n$. ConsequentlyThis gives strict inequality in Fatou lemma.