An extension of metric distributive lattices with an application in general analysis

Abstract

Introduction. This paper shows how to embed a metric distributive latlice(') [1, pp. 41 and 74] whose modular functional is bounded into a field of sets with a completely additive measure and applies this result to an abstract example of the second general analysis theory of E. H. Moore [10, vol. 1, pp. 14-15]. Moore devised the example as a generalization of Radon's generalization [11] of Hellinger integration [4]. Our investigation started with a lattice-theoretic interpretation of Moore's example. We were led to the extension problem in an effort to decide whether Moore's system was really more general than Radon's. Our result may be interpreted as stating that this is not the case, provided we assume bounded measure. We have been unable to answer this question in general. In connection with recent efforts to supply a very general basis for integration theory via lattice theory [2, 12, 3], it is interesting to point out that Moore's assumption of distributivity is essential in order that his basic matrix be positive. Even without the application we have in mind, our result on the extension of metric distributive lattices would seem to be interesting and important. The method of proof also has interest in that it combines the results of Stone [13] and of Wallman [14] with methods of MacNeille [8] and of Kakutani [6]. Our paper falls naturally into two parts. In Part I we present our process of extending a metric distributive lattice. Here we shall use the standard notations of theory and those special notations of lattice theory which are appropriate in distributive lattices and Boolean rings [1, pp. 96-97] with the following exception. We find occasion to employ a concept which, although used in many parts of mathematics, does not seem to have received a standard name. Roughly, this concept is that of a set of elements some of which are alike. Precisely, it is the concept of a maximal class of equivalent (non-null) sequences (paI aCA) of elements of a class $-=_3[p] where two sequences (pa CaCA) and (pI fCB) are equivalent if there is a one-to-one correspondence ac?z of A and B such that aT-f implies Pa = pp. The terminology