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

TL;DR

This paper investigates the topological properties of the space of probability measures on a compact space, establishing conditions under which these spaces inherit properties like being an ANR, Q-manifold, or Hilbert cube.

Contribution

It proves that the functor of probability measures preserves certain topological properties such as ANR, Q-manifold, and Hilbert cube structures under specific conditions.

Findings

Inclusion $P_f(X) ext{ is an ANR iff } X ext{ is an ANR}

The functor $P_f$ preserves Q-manifold and Hilbert cube properties

Properties of fibers of maps are preserved as ANR, Q-manifold, or Hilbert cube

Abstract

In this paper we prove that for a compact space $X$ inclusion $P_{f}(X)\in ANR$ 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 (the finite of Hilbert cube).