Perfect Probability Measures and Regular Conditional Probabilities

Abstract

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 $(X, \mathscr{S}, \mu)$ is a probability space and $\mu$ 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 $(X, \mathscr{S})$ having the property that every probability measure $\mu$ on $\mathscr{S}$ 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 $(X, \mathscr{S}, \mu)$ is a probability space, then $\mu$ is called perfect if for every $\mathscr{S}$-measurable function $f$ on $X$ and every set $A$ on the real line for which $f^{-1}(A)$ belongs to $\mathscr{S}$, there is a linear Borel set $B$ contained in $A$ such that $\mu(f^{-1}(A)) = \mu(f^{-1} (B))$. It is known (see [13]): (1) that a measure is perfect if and only if its restriction to every countably-generated sub-sigma-algebra is perfect, and (2) that the restriction to any sub-sigma-algebra of a perfect measure is perfect. Since the following characterization of perfectness will be used frequently in the sequel, we quote it here as a lemma. A proof may be found in [14]. Lemma 1. A measure $\mu$ on a measurable space $(X, \mathscr{S})$ is perfect if and only if for every $\mathscr{S}$-measurable function $f$ on $X$ there is a linear Borel set $B(f)$ contained in $f(X)$ such that $\mu(f^{-1}(B(f))) = \mu(X)$. The following definitions are those of Jirina [9]. Let $(X, \mathscr{S}, \mu)$ be a probability space and let $\mathscr{S}_1$ and $\mathscr{S}_2$ be sub-sigma-algebras of $\mathscr{S}$. Any function $\mu(\cdot, \cdot \mid \mathscr{S}_1, \mathscr{S}_2)$ defined on $\mathscr{S}_1 \times X$ will be called a conditional probability (c.p.) if it satisfies: CP1. for fixed $S$ in $\mathscr{S}_1, \mu(S, \cdot \mid \mathscr{S}_1, \mathscr{S}_2)$ is $\mathscr{S}_2$-measurable, and CP2. for every $S$ in $\mathscr{S}_1$, and every $T$ in $\mathscr{S}_2$, $\mu(S \cap T) = \int_T \mu(S_1, x \mid \mathscr{S}_1, \mathscr{S}_2) d(\mu \mid \mathscr{S}_2),$ where $(\mu \mid \mathscr{S}_2)$ is the restriction of $\mu$ to $\mathscr{S}_2$. If the c.p. $\mu(\cdot, \cdot \mid \mathscr{S}_1, \mathscr{S}_2)$ also satisfies: CP3. for each fixed $x$ in $X, \mu(\cdot, x \mid \mathscr{S}_1, \mathscr{S}_2)$ is a probability measure on $\mathscr{S}_1$, then it will be called a regular conditional probability (r.c.p.). It is well known that if $(X, \mathscr{S}, \mu)$ is a probability space, $\mathscr{S}_1$ is any sub-algebra of $\mathscr{S}$, and $\mathscr{S}_2$ is any sub-sigma-algebra of $\mathscr{S}$, then there is a c.p. $\mu(\cdot, \cdot \mid \mathscr{S}_1, \mathscr{S}_2)$. We shall denote by $\mu(\cdot, \cdot \mid \mathscr{S}, \mathscr{S}_2)$ any function satisfying CP1 and CP2, with $\mathscr{S}_1 = \mathscr{S}$.