Geometrical and topological properties of a subspace $P_f(X)$ of probability measures

We prove that for a compact $X$ the space $P_{f}(X)$ is an absolute neighbourhood retract if and only if $X$ is an absolute neighbourhood retract. Futher, we demonstrate that a functor $P_{f}$ preserves the property of a compact to be $Q$-manifold or Hilbert cube, preserves the propetry of a map to be absolute neighbourhood retract in a class of compact, to be $Q$-manifold or Hilbetr cube (finite sum of Hilbert cube).