On Conditional Probability Spaces Generated by a Dimensionally Ordered Set of Measures

Abstract

The concept of a “conditional probability space” is introduced, defined as the set $[S,\mathfrak{A},\mathfrak{B},{\bf P}]$, where S is an abstract space, $\mathfrak{A}$ is a $\sigma $-algebra of subsets of S, and $\mathfrak{B}$ is a nonempty subset of $\mathfrak{A}$; further ${\bf P}(A|B)$ is a function defined on $\mathfrak{A} \times \mathfrak{B}$, is a measure with respect to A, and satisfies certain conditions with respect to B. Every “probability space” generates a “conditional probability space” in a natural way; however, there are some “conditional probability spaces“ which are not generated by “probability spaces“. A method is given for constructing all “conditional probability spaces“.