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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On a weakly closed subset of the space of $ \tau $-smooth measures


Author: Wolfgang Grömig
Journal: Proc. Amer. Math. Soc. 43 (1974), 397-401
MSC: Primary 46E27
DOI: https://doi.org/10.1090/S0002-9939-1974-0338758-7
MathSciNet review: 0338758
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Abstract: It is known that a lot of topological properties devolve from a basic space $ X$ to the family $ {M_\tau }(X)$ of all $ \tau $-smooth Borel measures endowed with the weak topology (or to certain subspaces of $ {M_\tau }(X)$). The aim of this paper is to show that among these topological properties there cannot be properties which are hereditary on closed subsets but not on countable products of $ X$, e.g. normality, paracompactness, the Lindelöf property, local compactness and $ \sigma $-compactness. For this purpose it is proved that the countable product space $ {X^N}$ is homeomorphic to a closed subset of $ {M_\tau }(X)$. A further consequence of this result is for example that, for the family $ M_\tau ^1(X)$ of probability measures in $ {M_\tau }(X)$, compactness, local compactness and $ \sigma $-compactness are equivalent properties.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0338758-7
Keywords: $ \tau $-smooth measure, tight measure, weak convergence of measures, weak topology, normality, Lindelöf space, paracompactness, local compactness, $ \sigma $-compactness
Article copyright: © Copyright 1974 American Mathematical Society