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For nonnegative random variables $X_n$ and a sigma-algebra $\mathcal{G}$,almost surely. It follows from the ordinary Fatou lemma after testing both sides on every event in $\mathcal{G}$.

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

For $c>0$ and smooth compactly supported $u$, integration by parts givesDropping the nonnegative weighted derivative term proves the displayed bound. For a weak resolvent solution in the one-dimensional harmonic oscillator form domain, apply the bound to ${\chi }_{R}u$, where ${\chi }_R$ is a smooth cutoff. The cutoff error $-2{\chi }_R^{\prime }{u}^{\prime }-{\chi }_R^{\prime \prime }u$ is bounded in L2 space and tends to zero. The Fatou lemma then shows that $u^{\prime \prime }$ and $x^{2}u$ belong to L2 space, identifying the strong operator domain.

On a finite measure space, the convergence in the Birkhoff ergodic theorem also holds in $L^1$. Prove it first for bounded truncations by the dominated convergence theorem, then use the $L^1$ contraction $\Vert A_{n}h{\Vert }_1\le \Vert h{\Vert }_1$ and Fatou lemma to remove the truncation.

If $u_n\rightharpoonup u$ in $H^1({\mathbb{R}}^d)$ and $F:\mathbb{R}\to [0,\infty )$ is continuous, then $\int F(u)\le {\mathrm{liminf}}_n\int F(u_n)$. Start with a subsequence attaining the lower limit. The Rellich-Kondrachov compactness theorem gives strong local L2 space convergence along a diagonal subsequence, and a further subsequence converges almost everywhere. Its limit is $u$, by uniqueness of the local weak limit. Now apply the Fatou lemma. Nonnegativity is essential to this argument; convexity of $F$ is unnecessary. In particular $F(s)=1-\cos s$ is allowed despite its failure to be convex.

The Fatou lemma makes every time value of a nonnegative discrete-time local martingale integrable, assuming the usual integrable initial value. The integrable discrete-time result then makes it a true martingale.