Choquet theorem

Every point of a compact metrizable convex subset of a Hausdorff locally convex space is the barycenter of a Borel probability measure concentrated on its extreme points. The theorem is an existence assertion; uniqueness requires additional hypotheses such as a simplex structure.

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.

Maximize the integral of a continuous strictly convex function among probability measures with a fixed barycenter. The supporting measure lemma for affine upper envelopes implies that the maximizing measure has zero integral of the nonnegative envelope gap. Strict convexity makes this gap positive at every nonextreme point, so the measure is concentrated on the extreme boundary. Metrizability makes that boundary Borel and supplies the continuous strictly convex function.