Theorem 2

Tag 08P8, permanent. Cite it as [Bourbaki, Tag 08P8].

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

If $f_1$ and $f_2$ are two functions in $\mathscr{I}_+$ then

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

As $\varphi_1$ (resp. $\varphi_2$ ) runs over the set of functions in $\mathscr{K}_+$ such that $\varphi_1 \leq f_1$ (resp. $\varphi_2 \leq f_2$ ), the functions $\varphi_1 + \varphi_2$ form a directed set (for $\leq$ ) whose upper envelope is $f_1 + f_2$ . Therefore, by Th. 1,

\mu^*(f_1 + f_2) = \sup \mu(\varphi_1 + \varphi_2) = \sup (\mu(\varphi_1) + \mu(\varphi_2)),

where $(\varphi_1, \varphi_2)$ runs over the set of pairs of functions in $\mathscr{K}_+$ such that $\varphi_1 \leq f_1$ and $\varphi_2 \leq f_2$ ; since

\sup (\mu(\varphi_1) + \mu(\varphi_2)) = \sup \mu(\varphi_1) + \sup \mu(\varphi_2)

(GT, IV, §5, No. 7, Cor. 2 of Prop. 12), the theorem is proved.

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

Theorem 2

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