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In this paper we prove that for a compact space $X$ inclusion $P_{f}(X)\inANR$ holds if and only if $X\in ANR$. Further, it is shown that the functor$P_{f}$ preserves property of a compact to be $Q$-manifold or a Hilbert cube,properties of maps fibres to be $ANR$-compact, $Q$-manifold, Hilbert cube (thefinite of Hilbert cube).

Geometrical properties of the space $P_f(X)$ of probability measures