PERFECT PROBABILITY MEASURES AND REGULAR CONDITIONAL PROBABILITIES
1. Summary and introduction. In an attempt to refine the axiomatic model of a probability space introduced by Kolmogorov [11], Gnendenko and Kolmogorov [5] introduced the concept of a perfect probability measure. The desirability of some sort of refinement has been pointed out by several well known examples [2], [3], [8], which display a certain amount of pathology inherent in Kolmogorov's theory. It is known that if (X, 8, ,u) is a probability space and ,u is perfect, then each of the examples mentioned is ruled out. There have been attempts (e.g., [1], [10]) at characterizing all those measurable spaces (X, 8) having the property that every probability measure ,u on 8 is perfect. There have also been investigations [12], [13], [14] of the measure-theoretic properties of perfect measures and their relationships to other concepts in measure theory and probability theory. In this paper, we consider mixtures of perfect probability measures and their relationship to regular conditional probabilities. In Section 2, mixtures are defined and some interesting special cases are considered. Section 3 is a brief study of the imbedding of mixtures in a regular conditional probability space. Using known results and some results from Section 2, the perfectness of the underlying probability measure of a regular conditional probability space is characterized. Unless explicit mention is made to the contrary, the notation and terminology of [6] will be used throughout; however, the word "function," used without qualification, will always mean "real-valued function." If (X, 8, ,u) is a probability space, then ,u is called perfect if for every 8-measurable function f on X