  This paper deals with three major types of convergence of probability
measures on metric spaces: weak convergence, setwise converges, and convergence
in the total variation. First, it describes and compares necessary and
sufficient conditions for these types of convergence, some of which are
well-known, in terms of convergence of probabilities of open and closed sets
and, for the probabilities on the real line, in terms of convergence of
distribution functions. Second, it provides % convenient criteria for weak and
setwise convergence of probability measures and continuity of stochastic
kernels in terms of convergence of probabilities defined on the base of the
topology generated by the metric. Third, it provides applications to control of
Partially Observable Markov Decision Processes and, in particular, to Markov
Decision Models with incomplete information.
