A criterion for the uniform distribution of sequences in compact metric spaces
type: Item
instance of: scholarly article
title: A criterion for the uniform distribution of sequences in compact metric spaces
zbMATH Open document ID: 0655.10044
DOI: 10.1007/BF02843743
publication date: 1987-01-01T00:00:00Z
review text: The author generalizes a result of \textit{J. Horbowicz} [Indagationes Math. 43, 301-307 (1981; Zbl 0465.10039)] to compact metric probability spaces (X,\(\mu)\): Let f be a bounded measurable function such that f is \(\mu\)- almost everywhere continuous and different from zero. Then the sequence \((x_ n)\) in X is \(\mu\)-uniformly distributed if and only if \[ \lim_{N\to \infty}(1/N)\sum_{n\leq N, x_ n\in E}f(x_ n)=\int_{E}f(x) dx \] for every Jordan-measurable subset E of X (i.e. the boundary of E has measure zero). The result is generalized to more general summation methods and a situation in which \(\int_{E}f(x) dx\) is replaced by \(\int_{X}f(x)(Tg)(x) d\mu\), T: \(B\to C(X)\) a compact operator, such that C(X) is the closed linear span of T(B) and the constant functions.
Mathematics Subject Classification ID: 11K06
Mathematics Subject Classification ID: 54E45
zbMATH DE Number: 4069004
zbMATH Keywords: uniform distribution
zbMATH Keywords: Borel measure
zbMATH Keywords: compact metric probability spaces
zbMATH Keywords: bounded measurable function
zbMATH Keywords: summation methods
author: Robert F. Tichy
MaRDI profile type: Publication
cites work: Q3965144
cites work: Q3668337
cites work: Q5799972
cites work: Q4192121
cites work: Folgen auf kompakten Räumen
cites work: Zur formalen Theorie der Gleichverteilung in kompakten Gruppen
cites work: Q3916651
cites work: Q4767350
full work available at URL: https://doi.org/10.1007/bf02843743
OpenAlex ID: W2039702128
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 4069004