Metrics on ℕ and the Distribution of Sequences

Abstract

Abstract The aim of this paper is to study sequences of numbers as random variables. The asymptotic density will play the role of the probability. In the first part of this paper, the notion of natural metric on the set of natural numbers is defined. It is a metric so that the completion of ℕ is a compact metric space on which a probability Borel measure exists so that the sequence {n} is uniformly distributed. This condition connects the asymptotic density and the mentioned measure. A necessary and sufficient condition is derived so that a given metric is natural. Later, we study the properties of sequences uniformly continuous with respect to the given natural metric. Inter alia, the continuity ofdistribution function is characterized.