Chapter 7


The formulation of Kolmogorov’s conditional probability model is

abstractly discussed in this chapter, viewing it as a subclass of vector

measures on function spaces and the conditional expectation as a pro-

jection operator on the same spaces. Characterizations of both these

classes are presented for a family of function spaces, and their struc-

ture is thereby illuminated. In this context the Re´nyi (new) model is

compared with, and shown to be a particular case of, Kolmogorov’s

extended formulation. Vector integral representations of conditional

expectations and of measures as well as an application to a Gaussian

class is detailed. Finally a discussion of the relations between condi-

tioning and differentiation, complementing the work of Section 3.4, is

given and a general result on exact evaluations is included.