Compact Measures Have Loeb Preimages

Abstract

Abstract.A compact measure is a (possibly nontopological) measure that is inner-regular with respect to a compact family of measurable sets. The main result of this paper is that every compact probability measure is the image, under a measure-preserving transformation, of a Loeb probability space. This generalizes a well-known result about Radon topological probability measures. It is also proved that a compact probability space can be topologized in such a way that the measure is essentially Radon. IntroductionWhich probability measures can be represented by a Loeb space? In [2] Robert Anderson proved that every Radon probability measure on a Hausdorff space X is the image, under the standard part map, of a Loeb measure on *X. Depending on the topology on X, a converse can be proved; for example, D. Landers and L. Rogge [4] prove: if X is regular, then every probability measure, which is the image under the standard part map of a Loeb measure on *X, is Radon.The situation when X is not a Hausdorff topological space is more problematic, as the standard part map is not defined. An early approach to representing nontopological probability spaces by the Loeb measure [5] dispenses with measurable transformations altogether, but does not appear to have much application. The case when X is topological but not Hausdorff would seem to be easier than the general case, but still suffers from the lack of a standard part map (although T. Norberg [7] has recently constructed a natural analogue of the standard part map for so-called 'sober' spaces). Such spaces arise naturally in extremal theory and the theory of random closed sets (see e.g., [8] or [11]).In this paper I consider a nontopological analogue of a Radon space, called a compact probability space. The main theorem is that every compact probability space is the image, under a measure-preserving transformation cp , of a Loeb space, cp is very easy to construct, and acts very much like the standard part map; in particular, it is used to define a topology on X with respect to which cp is the standard part map, provided this topology is Hausdorff.

Compact measure spaces

“…We ought to remark that in both definitions here we are using the "modern" definitions, as in [8]. This note characterizes compact probability measures in terms of the representation of Boolean homomorphisms of their measure algebras, and shows that the same ideas can be used to give a direct proof of J. Pachl's theorem that any image measure of a countably compact measure is again countably compact.…”

“…We ought to remark that in both definitions here we are using the "modern" definitions, as in [8]. We ought to remark that in both definitions here we are using the "modern" definitions, as in [8].…”