I Representations of Points by Boundary Measures.- § 1. Distinguished Classes of Functions on a Compact Convex Set.- Classes of continuous and semicontinuous, affine and convex functions.— Uniform and pointwise approximation theorems.—Envelopes.—*Grothendieck’s completeness theorem.—Theorems of Banach-Dieudonné and Krein-Šmulyan*.- § 2. Weak Integrals, Moments and Barycenters.- Preliminaries and notations from integration theory.—An existence theorem for weak integrals.—Vague density of point-measures with prescribed barycenter.—*Choquet’s barycenter formula for affine Baire functions of first class, and a counterexample for affine functions of higher class*.- § 3. Comparison of Measures on a Compact Convex Set.- Ordering of measures.—The concept of dilation for simple measures.—The fundamental lemma on the existence of majorants.—Characterization of envelopes by integrals.—*Dilation of general measures.—Cartier’s Theorem*.- § 4. Choquet’s Theorem.- A characterization of extreme points by means of envelopes.—The concept of a boundary set.—Hervé’s theorem on the existence of a strictly convex function on a metrizable compact convex set.—The concept of a boundary measure, and Mokobodzki’s characterization of boundary measures.—The integral representation theorem of Choquet and Bishop — de Leeuw.—A maximum principle for superior limits of 1.s.c. convex functions.—Bishop — de Leeuw’s integral theorem relatively to a ?-field on the extreme boundary.—*A counterexample based on the “porcupine topology”*.- § 5. Abstract Boundaries Defined by Cones of Functions.- The concept of a Choquet boundary.—Bauer’s maximum principle.—The Choquet-Edwards theorem that Choquet boundaries are Baire spaces.—The concept of a Šilov boundary.—Integral representation by means of measures on the Choquet boundary.- § 6. Unilateral Representation Theorems with Application to Simplicial Boundary Measures.- Ordered convex compacts.—Existence of maximal extreme points.—Characterization of the set of maximal extreme points as a Choquet boundary.—Definition and basic properties of simplicial measures.—Existence of simplicial boundary measures, and the Caratheodory Theorem in ?n.—Decomposition of representing boundary measures into simplicial components.- II Structure of Compact Convex Sets.- § 1. Order-unit and Base-norm Spaces.- Basic properties of (Archimedean) order-unit spaces.—A representation theorem of Kadison.—The vector-lattice theorem of Stone-Kakutani-KreinYosida.—Duality of order-unit and base-norm spaces.- § 2. Elementary Embedding Theorems.- Representation of a closed subspace A of C?(X) as an A(K)-space by the canonical embedding of X in A*.—The concept of an “abstract compact convex” and its regular embedding in a locally convex Hausdorff space.—The connection between compact convex sets and locally compact cones.- § 3. Choquet Simplexes.- Riesz’ decomposition property and lattice cones.—Choquet’s uniqueness theorem.—Choquet-Meyer’s characterizations of simplexes by envelopes.—Edward’s separation theorem.—Continuous affine extensions of functions defined on compact subsets of the extreme boundary of a simplex.—Affine Borel extensions of functions defined on the extreme boundary of a simplex.—*Examples of “non-metrizable” pathologies in simplexes.*.- 4. Bauer Simplexes and the Dirichlet Problem of the Extreme Boundary.- Bauer’s characterizations of simplexes with closed extreme boundary.—The Dirichlet problem of the extreme boundary.—A criterion for the existence of continuous affine extensions of maps defined on extreme boundaries.- § 5. Order Ideals, Faces, and Parts.- Elementary properties of order ideals and faces.—Extension property and characteristic number.—Archimedean and strongly Archimedean ideals and faces.—Exposed and relatively exposed faces.—Specialization to simplexes.—The concept of a “part”, and an inequality of Harnack type.—Characterization of the parts of a simplex in terms of representing measures.—*An example of an Archimedean face which is not strongly Archimedean.*.- § 6. Split-faces and Facial Topology.- Definition and elementary properties of split faces.—Characterization of split faces by relativization of orthogonal measures.—An extension theorem for continuous affine functions defined on a split face.— The facial topology.—Specialization to simplexes.—*Near-lattice ideals, and primitive ideal space.—The connection between facial topology and hull kernel topology.—Compact convex sets with sufficiently many inner automorphisms.—A remark on the applications to C*-algebras.*.- § 7. The Concept of Center for A(K).- Extension of facially continuous functions.—The facial topology is Hausdorff for Bauer simplexes only.—The concept of center, and the connections with facially continuous functions and order-bounded operators.—Convex compact sets with trivial center.—*An example of a prime simplex.—Størmer’s characterization of Bauer simplexes.*.- § 8. Existence and Uniqueness of Maximal Central Measures Representing Points of an Arbitrary Compact Convex Set.- The relation xóy, and the concept of a primary point.—A point x is primary iff the local center at x is trivial.—The concept of a central measure.— Existence and uniqueness of maximal central measures in a special case.—The “lifting” technique.—Wils’ theorem on the existence and uniqueness of maximal central measures which are pseudo-carried by the set of primary points.- References.
