Proposition 3

Tag 08S7, permanent. Cite it as [Bourbaki, Tag 08S7].

Integration, chapter IV, § 4. Integrable functions and sets

— Let $\mathscr{B}$ be a filter base on $\mathscr{L}_F^1$ . Assume that there exists a compact set $K \subset X$ such that, for every set $M \in \mathscr{B}$ , all of the functions $f \in M$ have their support in $K$ . Under these conditions, if $\mathscr{B}$ converges uniformly on $X$ to $f_0$ , then the function $f_0$ is integrable and

\int f_0 \, d\mu = \lim_{\mathscr{B}} \int f \, d\mu .

For, $\mathscr{B}$ converges in mean to $f_0$ (§ 3, No. 3, Prop. 4).

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

Cited by

Example, § 4
Remark 2, § 4

Cites

§ 3, No. 3, Prop. 4