ABSTRACT
Chapter 7
ABSTRACTION OF KOLMOGOROV’S FORMULATION
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.
