On projections of finitely additive measures

Thomas Jech; Karel Prikry

doi:10.1090/S0002-9939-1979-0521891-9

On projections of finitely additive measures

Thomas Jech, Karel Prikry

Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

A theorem of Z. Frolik and M. E. Rudin states that for every two-valued measure μ on N, if F: N → N is such that F_* (μ) = μ 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 μ on N and a function F: N → N such that F_* (μ) = μ, and if A ⊆ N is such that F is one-to-one on A, then μ (A) ≤ 1/2.

Original language	English (US)
Pages (from-to)	161-165
Number of pages	5
Journal	Proceedings of the American Mathematical Society
Volume	74
Issue number	1
DOIs	https://doi.org/10.1090/S0002-9939-1979-0521891-9
State	Published - Apr 1979

Keywords

Finitely additive measure
Integral
Projection
Two-valued measure
Ultrafilter
Ultrafilter limit

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AB - A theorem of Z. Frolik and M. E. Rudin states that for every two-valued measure μ on N, if F: N → N is such that F* (μ) = μ 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 μ on N and a function F: N → N such that F* (μ) = μ, and if A ⊆ N is such that F is one-to-one on A, then μ (A) ≤ 1/2.

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On projections of finitely additive measures

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