There are Measurable Hamel Functions

Abstract

We say that a function $f:\mathbb{R}\to \mathbb{R}$ is a {\it Hamel function} if $f$, considered as a subset of $\mathbb{R}^2$, is a Hamel basis of $\mathbb{R}^2$. We show that there is a Marczewski measurable Hamel function. Additionally, we show that there is a Hamel function which is both Lebesgue measurable and with the Baire property.