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ArticleOriginal scientific text

Title

Vitali sets and Hamel bases that are Marczewski measurable

Authors 1, 2, 3

Affiliations

  1. Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Drive, Madison, WI 53706-1388, U.S.A.
  2. Institute of Mathematics, Bulgarian Academy of Sciences, Acad. G. Bontchev street, bl. 8, 1113 Sofia, Bulgaria
  3. Department of Mathematics, Auburn University, 218 Parker Hall, Auburn, AL 36849-5310, U.S.A.

Abstract

We give examples of a Vitali set and a Hamel basis which are Marczewski measurable and perfectly dense. The Vitali set example answers a question posed by Jack Brown. We also show there is a Marczewski null Hamel basis for the reals, although a Vitali set cannot be Marczewski null. The proof of the existence of a Marczewski null Hamel basis for the plane is easier than for the reals and we give it first. We show that there is no easy way to get a Marczewski null Hamel basis for the reals from one for the plane by showing that there is no one-to-one additive Borel map from the plane to the reals.

Bibliography

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Fundamenta Mathematicae
2000/166/3
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Pages:
269-279
Main language of publication
English
Received
1999-12-24
Accepted
2000-08-24
Published
2000
Exact and natural sciences