A short zero-one law proof of a result of Abian

Abstract

In this note a new and very short zero-one law proof of the following theorem of Abian is presented. The subset of the unit interval [0, 1) consisting of those real numbers whose Hamel expansions do not use a given basis element of a prescribed Hamel basis, has outer Lebesgue measure one and inner measure zero. Let {a, b, c, ...} be a Hamel basis for the real numbers. LetA be the subset of the unit interval [0, 1) consisting of those real numbers whose Hamel expansions do not use the basis elementa. Sierpinski [4, p. 108] has shown thatA is nonmeasurable in the sense of Lebesgue. Abian [1] has improved Sierpinski's result by showing thatm* (A), the outer measure ofA, is one and thatm * (A), the inner measure ofA, is zero. In this note a very short proof, using a zero-one law, of Abian's result will be presented. The following zero-one law is an immediate consequence of the Lebesgue Density Theorem [2, p. 290].