Ryll-Nardzewski fixed-point theorem

In functional analysis, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if

that is compact under the weak topology, then every group (or equivalently: every semigroup) of affine isometries of

[1] Later Namioka and Asplund [2] gave a proof based on a different approach.

Ryll-Nardzewski himself gave a complete proof in the original spirit.

[3] The Ryll-Nardzewski theorem yields the existence of a Haar measure on compact groups.