Vitali sets and Hamel bases that are Marczewski measurable
by Arnold W. Miller and Strashimir G. Popvassilev
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.
15 pages Latex 2e