Lemma 16.2.5 (Fatou, [Lemma 2.18, Fol99]). Let $(X, \cm, \mu)$ be a measure space, $\seq{f_n}\subset \mathcal{L}^{+}(X, \cm)$, then
\[\int \liminf_{n \to \infty}f_{n} d\mu \le \liminf_{n \to \infty}\int f_{n}d\mu\]
Proof. For each $n \in \natp$, $\inf_{k \ge n}f_{k} \le f_{n}$. By the monotone convergence theorem,
\[\int \liminf_{n \to \infty}f_{n} d\mu = \limv{n}\int \inf_{k \ge n}f_{k} d\mu \le \liminf_{n \to \infty}\int f_{n} d\mu\]
$\square$