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