On projections of finitely additive measures
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- by Thomas Jech and Karel Prikry PDF
- Proc. Amer. Math. Soc. 74 (1979), 161-165 Request permission
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 $\mu (A) \leqslant \tfrac {1}{2}$.References
- Zdeněk Frolík, Fixed points of maps of $\beta \,N$, Bull. Amer. Math. Soc. 74 (1968), 187–191. MR 222847, DOI 10.1090/S0002-9904-1968-11935-4
- Mary Ellen Rudin, Partial orders on the types in $\beta N$, Trans. Amer. Math. Soc. 155 (1971), 353–362. MR 273581, DOI 10.1090/S0002-9947-1971-0273581-5
- David Pincus and Robert M. Solovay, Definability of measures and ultrafilters, J. Symbolic Logic 42 (1977), no. 2, 179–190. MR 480028, DOI 10.2307/2272118
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 74 (1979), 161-165
- MSC: Primary 28A12; Secondary 54H10
- DOI: https://doi.org/10.1090/S0002-9939-1979-0521891-9
- MathSciNet review: 521891