Proposition 13

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

For every sequence $(f_n)$ of numerical functions $\geqslant 0$ defined on $X$ ,

\mu^*\left( \sum_{n=1}^\infty f_n \right) \leqslant \sum_{n=1}^\infty \mu^*(f_n).

It suffices to apply the relation (10) to the increasing sequence of functions $g_n = \sum_{k=1}^n f_k$ while taking into account that, by (9),

\mu^*(g_n) \leqslant \sum_{k=1}^n \mu^*(f_k).

In §5, No. 6, we will give conditions under which the two members of (12) are equal.

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

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