On measure and other properties of a Hamel basis

Abstract

F. B. Jones has established [2 ] several interesting measure-related properties of a Hamel basis. In particular, he established the existence of a Hamel basis which contains a nonempty perfect set (and, hence, supports a nontrivial Borel measure) and commented on the plausibility of the notion that a Hamel basis should be in some sense thick. Our purpose is to complement Jones' results by showing, subject to the continuum hypothesis which we assume throughout, that there exists a Hamel basis which intersects each first category set in, at most, a countable set and, hence, has universal measure zero (cf. [1]). It then follows, for instance, that the sum E+F= {e+f; eCE, f E F } of a universal null set E and a universal null set F need not be a universal null set and, moreover, an iterate En of E need not be Lebesgue measurable (cf. [2]). In order to fulfill our purpose it suffices to establish the following theorem. (We wish to acknowledge collaboration with R. E. Zink on problems related to the content of this note.)