  This paper describes sufficient conditions for the existence of optimal
policies for Partially Observable Markov Decision Processes (POMDPs) with Borel
state, observation, and action sets and with the expected total costs. Action
sets may not be compact and one-step cost functions may be unbounded. The
introduced conditions are also sufficient for the validity of optimality
equations, semi-continuity of value functions, and convergence of value
iterations to optimal values. Since POMDPs can be reduced to Completely
Observable Markov Decision Processes (COMDPs), whose states are posterior state
distributions, this paper focuses on the validity of the above mentioned
optimality properties for COMDPs. The central question is whether transition
probabilities for a COMDP are weakly continuous. We introduce sufficient
conditions for this and show that the transition probabilities for a COMDP are
weakly continuous, if transition probabilities of the underlying Markov
Decision Process are weakly continuous and observation probabilities for the
POMDP are continuous in the total variation. Moreover, the continuity in the
total variation of the observation probabilities cannot be weakened to setwise
continuity. The results are illustrated with counterexamples and examples.
