Proposition 12

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

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

— *If $f_1$ and $f_2$ are two numerical functions $\geqslant 0$ defined on X, then

\mu^*(f_1 + f_2) \leqslant \mu^*(f_1) + \mu^*(f_2).

For every function $h_1 \in \mathscr{I}_+$ such that $f_1 \leqslant h_1$ and every function $h_2 \in \mathscr{I}_+$ such that $f_2 \leqslant h_2$ , we have, by Th. 2,

\mu^*(f_1 + f_2) \leqslant \mu^*(h_1 + h_2) = \mu^*(h_1) + \mu^*(h_2),

whence (GT, IV, §5, No. 7, Cor. 2 of Prop. 12)

\mu^*(f_1 + f_2) \leqslant \inf_{h_1 \geqslant f_1, h_1 \in \mathscr{I}_+} \mu^*(h_1) + \inf_{h_2 \geqslant f_2, h_2 \in \mathscr{I}_+} \mu^*(h_2),

which is none other than the inequality (9).

Props. 10, 11 and 12 express that $\mu^*$ is an increasing, positively homogeneous and convex function on the set of numerical functions $\geqslant 0$ defined on X (Ch. I, No. 1). Note that if $f_1$ and $f_2$ are any two positive functions, the two members of (9) are not necessarily equal (§ 4, Exer. 8 d)); in §5, No. 6 we shall give conditions under which equality holds.

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

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Theorem 2, § 1

Proposition 12

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