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Marczewski measurable Vitali sets and Hamel bases
by
Strashimir G. Popvassilev
Auburn Univ., AL, and Inst. Math. Bulgarian Acad. Sci.
Coauthors: Arnold W. Miller (University of Wisconsin-Madison)
A subset M of a complete separable metric space X is called Marczewski null (resp., Marczewski measurable) if every perfect set P in X contains a perfect subset Q such that Q misses M (resp., either Q misses M or Q is contained in M).
Theorem. If X is a complete separable metric space and E is a Borel equivalence on X then there is a Marczewski measurable set V that contains exactly one representative from each equivalence class.
In particular there is a Marczewski measurable Vitali set (a set of reals that contains exactly one representative from each equivalence class, where x and y are equivalent iff their difference is rational).
Theorem. There is a Marczewski null Hamel basis for the reals over the rationals.
Also there are Hamel bases for X over the rationals, where X is either Rn or R\omega (the product of finitely or countably many copies of the real line), or Q\omega (the product of countably many copies of the rationals). A Hamel basis is a basis of X as a vector space over the rationals.
A function f is additive is f(x+y) = f(x) + f(y) for all x, y.
Theorem. There is no Borel (or even Baire) 1-1 additive function of the following form for any n = 1, 2, 3, ...
1. f : Rn+1 --> Rn
2. f : Rn --> Q\omega or f : Rn --> Z\omega
3. f : Q\omega --> Rn or f : Z\omega --> Rn
(where Z is the set of integers).
Homepage of Strashimir G. Popvassilev
Date received: February 9, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cady-35.