Proposition 14 (Fatou’s lemma)

Tag 08PR, permanent. Cite it as [Bourbaki, Tag 08PR].

Integration, chapter IV, § 1. Upper integral of a positive function

— For every sequence $(f_n)$ of numerical functions $\geqslant 0$ ,

\mu^*\left( \liminf_{n \to \infty} f_n \right) \leqslant \liminf_{n \to \infty} \mu^*(f_n).

For every integer $n$ , set $g_n = \inf_{p \geqslant 0} f_{n+p}$ ; the sequence $(g_n)$ is increasing and $\liminf_{n \to \infty} f_n = \sup_n g_n$ , whence, by (10),

\mu^*(\liminf_{n \to \infty} f_n) = \sup_n \mu^*(g_n);

but since $g_n \leqslant f_{n+p}$ for $p \geqslant 0$ , we have $\mu^*(g_n) \leqslant \mu^*(f_{n+p})$ , whence $\mu^*(g_n) \leqslant \inf_{p \geqslant 0} \mu^*(f_{n+p})$ and finally

\mu^*\left( \liminf_{n \to \infty} f_n \right) \leqslant \sup_n \left( \inf_{p \geqslant 0} \mu^*(f_{n+p}) \right) = \liminf_{n \to \infty} \mu^*(f_n).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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