A Note on the Summation of Sequences of 0's and 1's

Abstract

Any infinite sequence of O's and l's determines a dyadic fraction, and so corresponds with a point of the interval (0, 1). J. D. Hill (1, p. 558) showed that the set of points corresponding with the set of 0, 1 sequences which are summed by a given real T-matrix A is of Lebesgue measure 1 or 0. If the measure is 1, A is said to have the Borel property. Hill gave as a necessary condition for the real T-matrix A to have the Borel property that (1, p. 560)