METRICS ON N AND THE DISTRIBUTION OF SEQUENCES

Abstract

In the first part of this paper, the notion of natural metric on the set of natural numbers is defined. It is metric that the completion of $\mathbb{N}$ is a compact metric space that a probability Borel measure exists in
order that the sequence $\{n\}$ is uniformly distributed. A necessary and sufficient condition is derived that a given metric is natural. Later we study
the properties of sequences uniformly continuous with respect to the given
natural metric. Inter alia Theorems 5 and 8 characterize the continuity of
distribution function.