ABSTRACT
Although the general concept of conditioning of the preceding chap-
ter is abstractly shown to be well defined and has all the desirable
properties, actual computation of conditional probabilities satisfying
the Kolmogorov definition is a nontrivial task. In this chapter it is
shown that, in cases when the conditioning event has probability zero,
there are real problems of well posedness to use the L’Hoˆpital rule of
evaluation, and the latter method leads to surprising paradoxes. These
come from natural applications and are exemplified with works of Borel,
and Kac and Slepian. Methods of resolution of these difficulties and the
reasons for such ambiguities in the first place as well as the appropriate
procedures for correct solutions through the differentiation theory are
presented here in considerable detail. For a class of regular conditional
probabilities, unambiguous computational methods can be given and
these will be presented in Chapter 5 where a thorough discussion of
regularity is undertaken.
