The Choquet-Bishop-de Leeuw existence theorem

Abstract

Suppose that X is a nonmetrizable compact convex subset of a locally convex space E. As shown by examples in Bishop-de Leeuw [9], the extreme points of X need not form a Borel set. Thus, the statement “the probability measure μ is supported by the extreme points of X” is meaningless under our present definitions. There are at least two ways to get around this. We can drop the requirement that μ be a Borel measure (i.e., allow measures defined on a different σ-ring), or we can change the definition of “supported by” for Borel measures. An alternative definition might require that μ vanish on every Borel set which is disjoint from the set of extreme points, but Bishop and de Leeuw have shown that it is not always possible to obtain representing measures μ with this property. If, however, one demands only that μ vanish on the Baire subsets of X which contain no extreme points, then a representation theorem can be obtained. (Recall that the Baire sets are the members of the σ-ring generated by the compact G _δ sets.) Furthermore, this result leads easily to an equivalent theorem in which the definition of “supported by” remains formally the same, but the measure is no longer a Borel measure.