Reverse Fatou lemma. Let \((X, \mathscr{A}, \mu)\) be a measure space and \(\left(u_{n}ight)_{n \in \mathbb{N}} \subset \mathcal{M}^{+}(\mathscr{A})\). If \(u_{n} \leqslant u\) for all \(n \in \mathbb{N}\) and some \(u \in \mathcal{M}^{+}(\mathscr{A})\) with \(\int u d \mu<\infty\), then

\[
\limsup _{n ightarrow \infty} \int u_{n} d \mu \leqslant \int \limsup _{n ightarrow \infty} u_{n} d \mu
\]