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) |
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Pages (from-to) | 161-165 |
Number of pages | 5 |
Journal | Proceedings of the American Mathematical Society |
Volume | 74 |
Issue number | 1 |
DOIs | |
State | Published - Apr 1979 |
Keywords
- Finitely additive measure
- Integral
- Projection
- Two-valued measure
- Ultrafilter
- Ultrafilter limit