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Here the set is a nonempty compact metrizable convex subset of a Hausdorff locally convex space. The ambient space can be infinite-dimensional. Continuous affine functions separate its points, and weak-star compact dual balls with separable preduals provide examples. This general setting is needed for Choquet theorem.

For a real function in the $L^{\infty }$ unit ball, push uniform measure on $t\in [0,1]$ forward by the displayed sign-valued functions. In the weak-star topology this is a Borel probability concentrated on the extreme points, and Fubini's theorem shows that its barycenter is $f$. Weak-star compactness and separability of the predual $L^1$ place the example within Choquet theorem.