The Existence of Well Distributed Sequences in Compact Spaces

Baayen, P.C.; Hedrlín, Z.

doi:10.1016/s1385-7258(65)50027-5

Cited by 10 publications

(2 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…From now on X will be a compact Hausdorff space and I(X) will be the set of all continuous functions from X to the closed unit interval [0,1]. v will stand for any regular positive Borel measure of norm 1 (v(X) = 1).…”

Section: Notation and Terminologymentioning

confidence: 99%

On the equivalence of two modes of distribution of a sequence

Zame

1968

Monatshefte f�r Mathematik

View full text Add to dashboard Cite

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.

show abstract

Section: Notation and Terminologymentioning

confidence: 99%

On the equivalence of two modes of distribution of a sequence

Zame

1968

Monatshefte f�r Mathematik

View full text Add to dashboard Cite

show abstract

“…Let X be a compact Hausdorff space, let p be a regular normed Borel measure on X, and let C(X) be German: ,,gleichm~ /~-gleichverteih") if for every f E C(X) and for every real number e > 0 there exists an integer N(/, e), indepedent of a, such that (1) e t au and/or all a E/~ (HLAwKA [10]). …”

Section: Introductionmentioning

confidence: 99%