JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics. 1. Introduction. The density of a set of integers A = {a.} is usually defined as D (A) =-lim n/a. This function has some of the properties of a finitely additive measure on the countable space composed of the positive integers, although it is true that sets A and B may have a density while A u B does not. However, it is clear that generalized definitions of density can be given which apply to all sets of integers, and which are in fact true measures. [1, 231] [4].The present paper is largely devoted to an analysis of the measure defined on the set of positive integers by applying the Caratheodory extension to a simple basic measure; connection with the theory of Jordan content is very close. This also provides a simple model for classical measure theory; since points are to have zero measure, while the space on which the measure is defined is only countable, we must require only finite additivity. It is also clear that in studying the set of integers, we are studying any countable d iscrete space, for such a space can be mapped onto I.In Section 2, we construct the measure u and the class !)u of measurable sets. In Section 3, we prove measurability of certain special sets using number theoretic methods. We discuss in Section 4 certain questions related to sequences of sets and prove that the range of / is precisely the closed unit interval. Section 5 is devoted to an analysis of what we have called 'quasiprogressions.' It is proved that if a is irrational, the set of integers of the form [an + /3] intersects every arithmetic progression in an infinite set. In a sense, this is dual to the fact that if a is irrational, the fractional parts of an + ,3 are everywhere dense in the unit interval. In Section 6, we discuss a number of properties of ordinary density, and in the following section examine certain of its generalizations, using results dealing with regular summability methods. The concluding section deals with the customary dyadic mapping and questions of the relative measure of classes of measurable or densable sets. Several problems are left open. 2. Measure density. Let I denote the set of all positive integers. On the class of arithmetic progressions, we have what we may call a natural * Reeeived July 1, 1946.
