  This paper investigates natural conditions for the existence of optimal
policies for a Markov decision process with incomplete information (MDPII) and
with expected total costs. The MDPII is the classic model of a controlled
stochastic process with incomplete state observations which is more general
than Partially Observable Markov Decision Processes (POMDPs). For MDPIIs we
introduce the notion of a semi-uniform Feller transition probability, which is
stronger than the notion of a weakly continuous transition probability. We show
that an MDPII has a semi-uniform Feller transition probability if and only if
the corresponding belief MDP also has a semi-uniform Feller transition
probability. This fact has several corollaries. In particular, it provides new
and implies all known sufficient conditions for the existence of optimal
policies for POMDPs with expected total costs
