Proposition 1

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

Integration, chapter IV, § 7. Barycenters

Let E be a Hausdorff locally convex space, K a compact subset of E, and C the closed convex envelope of K in E. The set C then consists of the points of E that are barycenters of at least one positive measure of mass 1 on K.

This is nothing more than Prop. 5 of Ch. III, §3, No. 2 applied to the canonical injection of K into E.

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

Cited by

Lemma 2, § 7
Corollary, § 7
Proposition 3, § 7
Proposition 7, § 7

Cites

III, §3, No. 2
Prop. 5