Some Density Properties of Point Sets

Abstract

Introduction. Soon after his initial work on measure of linear point sets, Lebesgue [8] proved that at almost all points of a measurable linear set the density of the set exists and is unity, while at almost all points of the complement of the set the density exists and is zero. By using outer Lebesgue measure for linear sets which are not measurable, Blumberg [2] showed that the outer density of a linear set exists and is unity at almost all points of the set. Burkill and Haslam-Jones [31 showed that at almost all points of the complement of any linear set the outer density of the set exists and is either zero or unity. With the introduction of Carath6odory's definition of the p-dimensional measure of a set in q-dimensional space [4], new questions concerning density arose. In this paper we investigate some of these questions. Our demonstrations are all made for sets lying in the plane and having Carath6odory outer linear measure finite. The methods we use, however, are applicable with very little change to sets having outer p-dimensional measure finite and lying in a Euclidean space of q dimensions. By approaching the problem of density in this more general way we have obtained, as special cases of some of our results, all of the facts mentioned above for sets on a line. The nature of some of the further results of this paper, and their relation to the above theorems, may best be indicated by interpreting them also for the special case of sets on a line. With A any linear set, let C, denote the set of all points of the complement of A where the outer density of A exists and is unity. From Lebesgue's results, if A is measurable, C, has measure zero. We prove conversely, that if C, has measure zero, then A is measurable. Also regardless of whether A is measurable, we prove that the set A + C, is measurable and, moreover, has the same measure as the outer measure of A. Let C2 be the set of points of the complement of C, where the outer density of C, is unity. We show that C2 is a subset of A and also that A C2 is measurable with the same measure as the inner measure of A. We show also that the set C, + C2 is measurable and that its measure is the difference between the outer and inner measures of A.' The significance of some of our results is not, however, revealed by the special case of sets on a line. For example, if A is a subset of a linear set B, then at almost all points of A + C1, the outer densities of A and B both exist and are the same since they are both unity. If the assumption that the sets be linear is omitted, the outer densities of the sets need not even exist, i.e. each set may have