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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Borel measures and Hausdorff distance


Authors: Gerald Beer and Luzviminda Villar
Journal: Trans. Amer. Math. Soc. 307 (1988), 763-772
MSC: Primary 28C15; Secondary 54B20
MathSciNet review: 940226
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Abstract: In this article we study the restriction of Borel measures defined on a metric space $ X$ to the nonempty closed subsets $ \operatorname{CL} (X)$ of $ X$, topologized by Hausdorff distance. We show that a $ \sigma $-finite Radon measure is a Borel function on $ \operatorname{CL} (X)$, and characterize those $ X$ for which each outer regular Radon measure on $ X$ is semicontinuous when restricted to $ \operatorname{CL} (X)$. A number of density theorems for Radon measures are also presented.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0940226-0
PII: S 0002-9947(1988)0940226-0
Keywords: Borel measure, Radon measure, Hausdorff distance, free union of measures, UC space, Baire space, weakly Baire space, Vietoris topology
Article copyright: © Copyright 1988 American Mathematical Society