  This paper studies transition probabilities from a Borel subset of a Polish
space to a product of two Borel subsets of Polish spaces. For such transition
probabilities it introduces and studies the property of semi-uniform Feller
continuity. This paper provides several equivalent definitions of semi-uniform
Feller continuity and establishes its preservation under integration. The
motivation for this study came from the theory of Markov decision processes
with incomplete information, and this paper provides fundamental results useful
for this theory.
