Proposition 1

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

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

Let $X$ be a topological space and $G$ a solvable group. We assume that $G$ operates on $X$ on the left, in such a way that for all $g \in G$ , the mapping $x \mapsto g.x$ from $X$ into $X$ is continuous. Then there exists a mean on $X$ which is invariant under $G$ .

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

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