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Proceedings of the American Mathematical Society

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



On projections of finitely additive measures

Authors: Thomas Jech and Karel Prikry
Journal: Proc. Amer. Math. Soc. 74 (1979), 161-165
MSC: Primary 28A12; Secondary 54H10
MathSciNet review: 521891
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Abstract: A theorem of Z. Frolík and M. E. Rudin states that for every two-valued measure $ \mu $ on N, if $ F:N \to N$ is such that $ {F_ \ast }(\mu ) = \mu $ then $ F(x) = x$ for almost all x. We prove that a generalization of this theorem fails for measures in general: Theorem. There exist a translation invariant measure $ \mu $ on N and a function $ F:N \to N$ such that $ {F_ \ast }(\mu ) = \mu $, and if $ A \subseteq N$ is such that F is one-to-one on A, then $ \mu (A) \leqslant \tfrac{1}{2}$.

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Keywords: Finitely additive measure, two-valued measure, ultrafilter, projection, ultrafilter limit, integral
Article copyright: © Copyright 1979 American Mathematical Society

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