Corollary 4

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

Integration, IV, p. 73, § 5. Measurable functions and sets

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

\int^*\left( \sum_n f_n \right) d|\mu| = \sum_n \int^* f_n d|\mu|.

In view of §1, No. 3, Th. 3, we are reduced to proving that for any two measurable functions $f \geqslant 0, g \geqslant 0$ on $X$ ,

\int^*(f + g) d|\mu| = \int^* f d|\mu| + \int^* g d|\mu|.

This is nothing more than the additivity of the integral when $f$ and $g$ are integrable. In the contrary case, if for example $f$ is not integrable, then $\int^* f d|\mu| = +\infty$ by Th. 5; a fortiori, $\int^*(f + g) d|\mu| = +\infty$ .

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

Cited by

Corollary 5, § 5
Proposition 2, § 1

Cites

§1, No. 3, Th. 3
Th. 5