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

Summary 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].