SOME RESULTS CONCERNING HAMEL BASES

Abstract

In this paper we show that under Martin’s Axiom there is a Hamel basis $H$ for which the set $E^{+}(H)$ of all finite linear combinations of elements from $H$ with nonnegative rational coefficients is both of Lebesgue measure zero and of the first Baire category. We also obtain that the additive group of the reals $ℝ$ is a direct sum of two subgroups one of which is uncountable, of measure zero and of the first Baire category.