Corollary 2

Tag 08N3, permanent. Cite it as [Bourbaki, Tag 08N3].

Integration, chapter III, § 2. Support of a measure

— Let $\mu$ be a positive measure on $X$ ; if $f \in \mathscr{K}(X; \mathbf{C})$ is such that $f(x) \geq 0$ on $\operatorname{Supp}(\mu)$ , then $\mu(f) \geq 0$ .

For, $f = |f|$ on $\operatorname{Supp}(\mu)$ , therefore $\mu(f) = \mu(|f|) \geq 0$ by Corollary 1.

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

Cited by

Proposition 11, § 2
Corollary 3, § 2
Corollary 3, § 2
Proposition 4, § 3

Cites

Corollary 1