On Uniform Distributions on Metric Compacta

Abstract

We introduce the notion of uniform distribution on a metric compactum. The desired distribution is defined as the limit of a sequence of the classical uniform distributions on finite sets which are uniformly distributed on the compactum in the geometric sense. We show that a uniform distribution exists on the metrically homogeneous compacta and the canonically closed subsets of a Euclidean space whose boundary has Lebesgue measure zero. If a compactum (satisfying some metric constraints) admits a uniform distribution then so does its every canonically closed subset that has zero uniform measure of the boundary. We prove that compacta, admitting a uniform distribution, are dimensionally homogeneous in the sense of box-dimension.