chapter  5
58 Pages

REGULARITY OF CONDITIONAL MEASURES

REGULARITY OF CONDITIONAL MEASURES

In this chapter various restrictions on the underlying probability

space are given for the (Kolmogorov) conditional probability functions

to behave like scalar or ordinary measures. Existence results for such

measures, some easily recognizable sufficient conditions for the pur-

pose, and then the related problem of disintegrating a measure as a

“convex combination” of regular conditional measures are treated in

detail. The latter results have a close relationship with the topology

of the underlying space and this is discussed. These existence results

are fundamental for theoretical developments. However, no algorithm

is available for their constructions in general. But, for a broad class

of regular conditional probabilities (and expectations) we include, in

the last two sections, constructive computational methods with illus-

trations to (Brownian motion) processes and functionals on them.