A new setting: The Choquet boundary

Abstract

In the Introduction, the Riesz representation theorem was reformulated as a representation theorem of the Choquet type. Although the conclusion of the Riesz theorem is quite sharp (for each element of the convex set X under consideration there exists a unique representing measure supported by ex X), the hypotheses restrict its application to a very special class of compact convex sets. In what follows we will (among other things) describe a related family of sets which appears to be only slightly larger than that involved in the Riesz theorem, but which actually “contains” all the sets which interest us, in the sense that every compact convex subset of a locally convex space is affinely homeomorphic to a member of the family.