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Measures on hyperspaces


Authors: Włodzimierz J. Charatonik and Matt Insall
Journal: Proc. Amer. Math. Soc. 144 (2016), 4753-4757
MSC (2010): Primary 28E99, 54B20
DOI: https://doi.org/10.1090/proc/13215
Published electronically: July 26, 2016
MathSciNet review: 3544527
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Abstract: From a measure space, $ (X, \mu ^{\mathbb{X}})$ we define a measure $ \mu ^{\mathbb{P}(\mathbb{X})}$ on the power set of $ X$. If $ (X, \tau )$ is a compactum, whose topology $ \tau $ is compatible with the measure $ \mu ^{\mathbb{X}}$ on $ X$, then the measure $ \mu ^{\mathbb{P}(\mathbb{X})}$ restricts to a natural measure on the hyperspace of closed sets of that given compactum. Surprisingly, under very mild conditions, $ \mu ^{\mathbb{P}(\mathbb{X})}$ is always supported on the hyperspace of finite subsets.


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Additional Information

Włodzimierz J. Charatonik
Affiliation: Department of Mathematics and Statistics, Missouri University of Science and Technology, 400 W. $12^{th}$ Street, Rolla, Missouri 65409-0020
Email: wjcharat@mst.edu

Matt Insall
Affiliation: Department of Mathematics and Statistics, Missouri University of Science and Technology, 400 W. $12^{th}$ Street, Rolla, Missouri 65409-0020
Email: insall@mst.edu

DOI: https://doi.org/10.1090/proc/13215
Keywords: Hyperspace, measure, topological space
Received by editor(s): February 11, 2015
Received by editor(s) in revised form: November 30, 2015
Published electronically: July 26, 2016
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2016 American Mathematical Society