Purification and saturation

Abstract

This paper illustrates the general technique established in 1984 by Hoover and Keisler for extending certain types of results from atomless Loeb measure spaces to measure spaces that we shall call "nowhere countably generated". The Hoover-Keisler technique is applied here to further extend the authors' 2006 generalization of a theorem of Dvoretzky, Wald and Wolfowitz on the purification of measure-valued maps. The authors' 2006 result was first extended to these more general spaces by K. Podczeck in 2007; he used new results in functional analysis produced for that purpose. This paper demonstrates that, in general, such extensions follow from the Hoover-Keisler technique. Moreover, adaptations of counterexamples from earlier papers show that the extension obtained here holds only for nowhere countably generated spaces. © 2009 American Mathematical Society.