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

 

On perfect measures


Author: G. Koumoullis
Journal: Trans. Amer. Math. Soc. 264 (1981), 521-537
MSC: Primary 28C15; Secondary 03E55, 54D18, 60A99
MathSciNet review: 603778
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Abstract: Let $ \mu $ be a nonzero positive perfect measure on a $ \sigma $-algebra of subsets of a set $ X$. It is proved that if $ \{ {A_i}:i \in I\} $ is a partition of $ X$ with $ {\mu ^ \ast }({A_i}) = 0$ for all $ i \in I$ and the cardinal of $ I$ non-(Ulam-) measurable, then there is $ J \subset I$ such that $ { \cup _{_{i \in J}}}{A_i}$ is not $ \mu $-measurable, generalizing a theorem of Solovay about the Lebesgue measure. This result is used for the study of perfect measures on topological spaces. It is proved that every perfect Borel measure on a metric space is tight if and only if the cardinal of the space is nonmeasurable. The same result is extended to some nonmetric spaces and the relation between perfectness and other smoothness properties of measures on topological spaces is investigated.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0603778-X
PII: S 0002-9947(1981)0603778-X
Keywords: Perfect, $ \sigma $-additive, $ \tau $-additive, tight measure, measurable cardinal, weakly $ \theta $-refinable, weakly metacompact, developable, realcompact space
Article copyright: © Copyright 1981 American Mathematical Society