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Probability measure functors preserving the ANR-property of metric spaces. (English) Zbl 0719.54023

For a metric space X let \(P_{\infty}(X)\) be the space of all probability measures on X with finite supports equipped with the weak topology induced by the Set of all bounded continuous functions on X. Every \(\mu \in P_{\infty}(X)\) can be written in the form \(\mu =\sum^{k}_{i=1}m_ i\delta_{x_ i}\), where \(m_ i\in (0,1)\) and \(\sum^{k}_{i=1}m_ i=1\). The support of \(\mu\) is the set \(\{x_ 1,...,x_ k\}\). Let \(P_ n(X)\) be the space \(\{\mu \in P_{\infty}(X):\) supp \(\mu\) consists of no more than n points\(\}\). V. V. Fedorchuk [Sov. Math. Dokl. 22, 849-853 (1980; translation from Dokl. Akad. Nauk SSSR 255, 1329-1333 (1980; Zbl 0505.54033)] proved that if X is a compact ANR-space then \(P_ k(X)\) is also ANR for every \(k\in N\). In the present paper the authors proved the same result without the compactness assumption on X. It is shown, as a corollary of this theorem, that \(P_ k(l_ 2)\) is homeomorphic to \(l_ 2\) for every \(k\in N\).
Reviewer: V.Valov (Sofia)

MSC:

54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
60B05 Probability measures on topological spaces
58D15 Manifolds of mappings
58B05 Homotopy and topological questions for infinite-dimensional manifolds
57N20 Topology of infinite-dimensional manifolds
37-XX Dynamical systems and ergodic theory

Citations:

Zbl 0505.54033
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References:

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