ABSTRACT
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.
