Marczewski measurable Vitali sets and Hamel bases
by
Strashimir G. Popvassilev
Auburn University, AL, and Inst. Math. Bulg. Acad. Sci, Sofia, Bulgaria
Coauthors: Arnold W. Miller (University of Wisconsin-Madison)

A set M is in the class (s) of Marczewski measurable sets if every perfect set P contains a perfect set Q such that either Q is a subset of M or Q misses M. If every perfect set P contains a perfect set Q which misses M, then M is in the class (s^o) of Marczewski null sets.

Examples are given of a Vitali set and, under CH, a Hamel basis that are Marczewski measurable and perfectly dense in R. Under CH there is a Hamel basis which is a Lusin set (and, hence, Marczewski null set). In ZFC there is a Marczewski null Hamel basis for the plane as a vector space over the rationals. We might hope to prove the same result for the reals by getting a Borel linear isomorphism between the plane and the reals, but this will not work because any additive Borel map R ×R --> R fails to be one-to-one. There exists a Hamel basis for the reals which is a Marczewski null set, though the proof is a little messier than the one for the plane.

The Vitali set example answers a question posed by Prof. Jack Brown in his course Real Functions, read in Spring quarter'98 in Auburn University.

http://www.auburn.edu/~popvast/

Date received: June 4, 1999