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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On continuity of invariant measures


Author: Andrew Adler
Journal: Proc. Amer. Math. Soc. 41 (1973), 487-491
MSC: Primary 28A70; Secondary 43A07
MathSciNet review: 0328025
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Abstract: Main Theorem. Let $ \Phi $ be a set of transformations on a set $ X$. The following conditions are then equivalent:

(1) There is a noncontinuous finitely additive measure defined on all subsets of $ X$ and invariant under all transformations in $ \Phi $.

(2) There is an integer $ m$ such that for any finite subset $ F$ of $ \Phi $ there is a finite subset $ {A_F}$ of $ X$, with no more than $ m$ elements, such that each $ f$ in $ F$ acts as a permutation on $ {A_F}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0328025-9
PII: S 0002-9939(1973)0328025-9
Keywords: Finitely additive, invariant, measure, continuous, ultrafilter
Article copyright: © Copyright 1973 American Mathematical Society