Corollary 3

Tag 08M3, permanent. Cite it as [Bourbaki, Tag 08M3].

Integration, chapter III, § 1. Measures on a locally compact space

— If $X$ is compact, the set of positive measures $\mu$ on $X$ such that $\|\mu\| = 1$ is vaguely compact.

For, it is the intersection of the vaguely compact set (Cor. 2) of measures such that $\|\mu\| \leq 1$ and the vaguely closed sets defined respectively by the relations $\mu \geq 0$ and $\mu(1) = 1$ (No. 8, Cor. 2 of Prop. 10).

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

Cited by

Proposition 5, § 3

Cites

Prop. 10
Cor. 2