  The axiomatic foundation of probability theory presented by Kolmogorov has
been the basis of modern theory for probability and statistics. In certain
applications it is, however, necessary or convenient to allow improper
(unbounded) distributions, which is often done without a theoretical
foundation. The paper reviews a recent theory which includes improper
distributions, and which is related to Renyi's theory of conditional
probability spaces. It is in particular demonstrated how the theory leads to
simple explanations of apparent paradoxes known from the Bayesian literature.
Several examples from statistical practice with improper distributions are
discussed in light of the given theoretical results, which also include a
recent theory of convergence of proper distributions to improper ones.
