Fatou's lemma

For nonnegative measurable functions,

If $Y_n$ are random variables bounded above by an integrable function $Z$, with all relevant expectations well defined, Fatou's lemma applied to the nonnegative variables $Z-Y_n$ gives ${\mathrm{limsup}}_n\mathbb{E}{Y}_n\le \mathbb{E}{\mathrm{limsup}}_n{Y}_n$. In particular, bounded indicator random variables that tend almost surely to zero have probabilities tending to zero.

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.

For nonnegative numbers $a_{n,k}$, Fatou lemma applied to counting measure gives

SetThen $0\le g_n\uparrow {\mathrm{liminf}}_k{f}_k$. The monotone convergence theorem and $g_n\le f_k$ for every $k\ge n$ give

On $([0,1],\lambda )$, the functionsconverge pointwise to zero but satisfy $\int f_{n}d\lambda =1$. Hence the two sides of Fatou lemma can be zero and one.