Vol. 66, No. 1, 1976

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Choquet simplexes and σ-convex faces

Kenneth R. Goodearl

Vol. 66 (1976), No. 1, 119–124

The purpose of this paper is to present a simple characterization of the split-faces in a Choquet simplex K, i.e., those faces F such that K is a direct convex sum of F and its complementary face. It is shown that a face F is a split-face if and only if it is σ-convex, i.e., closed under infinite convex combinations. This is proved by means of a measure-theoretic characterization of the σ-convex faces of K. As a consequence, it is shown that the lattice of σ-convex faces of a Choquet simplex forms a complete Boolean algebra.

Mathematical Subject Classification 2000
Primary: 46A99
Received: 4 February 1976
Published: 1 September 1976
Kenneth R. Goodearl
University of California, Santa Barbara
Santa Barbara CA
United States