Compact metrizable convex set

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.

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.

Work with the real Lp space $L^{\infty }((0,1))$ for Lebesgue measure, so its predual $L^1((0,1))$ is separable. Choose a measurable representative $f$ of a unit-ball element with $-1\le f\le 1$ and putFor every $t$, $H_t$ has modulus one almost everywhere, so it is an extreme point of the real $L^{\infty }$ closed unit ball. The map $t\mapsto H_t$ is Borel for the weak-star topology: pairing with any $g\in L^1$ is measurable, and separability makes the weak-star unit ball metrizable. Push forward uniform probability measure on $[0,1]$ through this map.

For almost every $s$,Fubini's theorem then gives ${\int }_0^1\int g(s)H_t(s)dsdt=\int g(s)f(s)ds$ for every $g\in L^1$. Thus the resulting Borel probability measure is concentrated on the extreme points and has barycenter $f$.

For $f(s)=2s-1$, the profiles are the threshold steps $H_t(s)=21_{\{t\le s\}}-1$. Averaging these sign-valued extreme points produces the linear profile. This is an explicit Choquet theorem representation in a setting where the required metrizability is actually available.

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.

For a bounded real function on a compact convex set, this envelope is the infimum of its continuous affine majorants. It is finite, concave and upper semicontinuous, and dominates the function. For continuous functions it describes the maximal integral among probability measures with a specified barycenter. Its homogeneity, subadditivity and affine-translation identities give the supporting measure lemma for affine upper envelopes.

Let $\mu$ be a Borel probability measure on the compact convex set $K$, and fix $f\in C(K,\mathbb{R})$. the sublinear functional $p(g)=\mu (\overline{g})$ has a linear supporting functional taking the value $p(f)$ at a specified continuous $f$, by the Hahn-Banach theorem. It is positive and normalized because $p$ has these values on constants. The Riesz-Markov-Kakutani representation theorem gives the probability $\nu$. Testing affine functions and their negatives shows that $\mu$ and $\nu$ have the same barycenter.

The barycenter of a probability measure on a compact convex set is the point satisfying the displayed affine integral identities. It exists by approximating the measure by finitely supported measures and using compactness of their finite convex combinations. It is unique because continuous affine functions separate points. In a Banach-space setting it agrees with the appropriate vector integral when that integral is defined.