Integral representations by non-finite Radon measures and measures of finite variation

Abstract

While Choquet's representation theorems yield a natural and elegant approach to integral representations by probability measures, things become more complicated when unbounded Radon measures are involved. It is the aim of this note to remedy this disadvantage. Let X + {0} be a weakly complete cone in some locally convex Hausdorff space E. If H is a cap of X such that the cone generated by H is dense in X, then each point x E X is the resultant of a positive linear form on a suitable subspace ~ C ~(H\{0}) such that the continuous functions with compact support as well as the restrictions to H\{0} of continuous linear forms on E are contained in 6~. Moreover, each function f ~ 6 e is majorized and minorized by some continuous linear form on E. The convex members of ,5 e define an ordering <~ in the cone of all positive linear forms on 6 e similarly to the Choquet-Bishop-de Leeuw ordering. To each point x e X there exists a <~-maximal positive linear form # on 5 e with resultant x, and # is unique iff X is a lattice cone. Furthermore,/~ turns out to be a Radon measure whenever x is the supremum of an upward directed set in the cone generated by H. Thus, if H is an almost universal cap, each point x s X is representable by <~-maximal positive Radon measures. The <~-maximal measures are the complete analogues to the maximal probability measures in classical Choquet theory, and, in fact, generalize them. Applications to function algebras and integral representations of m-monotonic functions including the theorem of Bernstein and Widder for completely monotonic functions on open subcones of IR n demonstrate the efficiency of the general theory.