Hamel bases and measurability

Abstract

This is a note - set in the background of some historic comments - discussing the relationship between measurability and Hamel bases for R over Q. We explicitly note that such a basis must necessarily fail to be Borel measurable (or even 'analytic' in the sense of descriptive set theory). We also discuss some constructions in the literature which yield Hamel bases which even fail to be Lebesgue measurable, and discussan elementary construction of a Hamel basis which is Lebesgue measurable.