On Measure and Other Properties of a Hamel Basis

R. B. Darst
1965 Proceedings of the American Mathematical Society  
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
more » ... basis which intersects each first category set in, at most, a countable set and, hence, has universal measure zero (cf. [l]). It then follows, for instance, that the sum P + P= \e-\-f; eEE, fEF} 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.) Theorem. There exists a Hamel basis H which intersects each perfect nowhere dense set in, at most, a countable set. Before proceeding to a proof of the theorem we wish to state the following lemma which we shall have occasion to use. Lemma. If Q is a first category subset 0/ (0, l] and x is a point of (0, 1), then there exists a point y of (0, x) such that x+yEix, 1) and neither y nor x+y is an element of Q. Proof of Theorem. Let Í2 denote the first uncountable ordinal and let {Pa)"<í¡ and {xa}"<n, xi = l, be well orderings of the perfect nowhere dense subsets of (0, l] and the points of (0, 1]. We shall define H=\Ja<si Ha inductively as follows. Let Hi-{l}, Ri = 0. Suppose 1 <a<Q and, for 1 ^ß<a, Hß and Rß satisfy: (1) Hß is, at most, countable. (2) The elements of Hß are linearly independent over the rationale. (3) Rß is a subset of the linear span H$ of Hß. (4) HßC\Rß = 0.
doi:10.2307/2033895 fatcat:y2ulwmax5rhjhdbxa6h5ean5hq