Definition 1

Tag 07WK, permanent. Cite it as [Bourbaki, Tag 07WK].

Topological Vector Spaces, chapter IV, Appendix 0. Fixed points of groups of affine transformations

A positive linear form $\mu$ on the space $\mathscr{B}(X; \mathbf{R})$ , where X is a topological space, for which $\|\mu\| = 1$ , is called a mean on X.

\* When X is compact, a mean on X is a positive measure on X such that $\mu(X) = 1$ . \*

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

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