Vitali sets and Hamel bases that are Marczewski measurable

Volume 166 / 2000

Arnold W. Miller, Strashimir G. Popvassilev Fundamenta Mathematicae 166 (2000), 269-279 DOI: 10.4064/fm-166-3-269-279


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.


  • Arnold W. Miller
  • Strashimir G. Popvassilev

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