On the equivalence of two modes of distribution of a sequence

Abstract

O. Introduction. If I is the unit interval and # is ordinary Lebesgue measure then the concept of the uniform distribution of a sequence (x~), x~ e I, is well known. This notion can be extended quite naturally to the case of a compact Hausdorff space X with a regular Borel measure ~. In addition, we can study modes of distribution which are stronger than uniform distribution. In particular, a recent paper [2] by Baayen and Helmberg studies sequences which are uniformly distributed, well distributed, weakly well distributed, almost well distributed, almost well distributed I and almost well distributed//. (These terms will be defined in the next section). The sets of sequences which are so distributed will be denoted by U, W, WW, A W, AWI and A WII, respectively. Under the assumption that X satisfies the second axiom of countability (and hence, by Furthermore, if v~ represents the induced measure in the countably infinite product space generated by X, then we also have that v~ (AWI) : ] =-v~(WW),v~(AWII): 0. Baayen and Helmberg also prove that A W r WW, but they raise the question of the relationship between AWI and WW. In this paper we will see that if X satisfies the second axiom of countability and v is not a point measure then AWI WW.