On the problem of evanescent processes

Abstract

1. Let R denote the set of real numbers. Let G be a countable dense subgroup of R. We construct a nontrivial ca-finite measure m on R such that (i) m is nonatomic, i.e. ,u({x}) =0 for every real number x, (ii) m is singular with respect to the Lebesgue measure L on R, (iii) m is invariant under translation by members of G, i.e., m(A +g) =m(A) for all gG n==1, 2, * * * and let D be the set of all those numbers x in whose expansion an takes values 0 or 9. M\lore accurately D is the set of all numbers x in the unit interval I such that x can be expanded by using 0 and 9 alone. Geometrically D is the Cantor set obtained by deleting the middle 8/lOths. Step 2. Here we state a known result and indicate its proof. Let A5 denote the set of all those numbers in the unit interval I whose decimal expansion does not involve the number 5.