Trivial bundles of spaces of probability measures

It is proved that the probability measure functor $P$ carries open mappings $f\colon X\to Y$ of finite-dimensional compact metric spaces with infinite fibers $f^{-1}y$ into $Q$-bundles. If in addition the fibers $f^{-1}y$ do not have isolated points, then it is possible to drop the condition that $X$ be finite-dimensional. Also, necessary and sufficient conditions are given for the mapping $P(f)$ to be a trivial bundle with fiber homeomorphic to a Tychonoff cube in the case of a mapping $f$ onto a dyadic compactum.
Bibliography: 27 titles.