On problems of Von Neumann and Maharam

Abstract

Lattices are ubiquitous in mathematics. The beauty and simplicity of the abstraction and the ability to tie together seemingly unrelated pieces of mathematics are certainly appealing to mathematician-as-artist. The introduction of new and nontrivial techniques for the solution of outstanding problems, for example in measure theory, is mathematically rewarding; the discovery of new questions which become natural to ask in the context of lattice flavour in measure theory is undoubtedly vigorously intriguing. Lattice pervades all branches of human knowledge. Because of diagrammatic representation the very simplicity of the basic concepts and the degree of abstraction in its relationship to other branches, lattice theory has an intuitive and aesthetic appeal. Our effort here is to create relations between different lattices. We have considered some important class of lattices which are necessary understand this dissertation and are useful for applications in other branches of mathematics. The diagram depicts the inclusion relationships among some important subclasses of lattices. Our endeavour here is to investigate the weak distributivity of Boolean -algebras satisfying the countable chain condition (ccc). It addresses primarily the question when such algebras carry a -additive measure. We use as a starting point the year old problem of John von Neumann stated in 1937 in the Scottish Book. He posed the problem if the countable chain condition (ccc) and weak distributivity are sufficient for the existence of -additive measure. Subsequent research has shown that the problem has two aspects: one set theoretic and one combinatorial. Recent results provide a complete solution of both the set theoretic and combinatorial problems. Our effort here is to survey the history of von Neumann’s Problem and outline the solution of the set theoretic problem. The technique that we describe owes much to the early work of Dorothy Maharam. We investigate von Neumann’s question : whether every weakly distributive complete ccc Boolean algebra is a measure algebra. We shall present a number of additional necessary and sufficient conditions for a complete ccc Boolean algebra B to carry Maharam submeasure. It turns out that some of the properties are natural generalizations of the conditions of weakly distributive. We remark that the assumption of weak distributivity is not necessary in reformulation of the Housdorffness but is necessary in reformulation of the G property. Finally we shall show that strategic versions of weak distributivity are equivalent in carrying a Maharam submeasure. To make the idea clear we have considered an infinite game of two players. Players I and II take turns to successively produce two infinite sequences of moves. The games in terms of winning strategies that we have considered here are : weak distributivity game, diagonal game and bounding game. We also cite here some examples of weak distributivity complete ccc algebras that are not Maharam. We investigate the sequential topology s on a complete Boolean algebra B determined by algebraically convergent sequences in B. We show the role of weak distributivity of B in separation axioms for sequential topology. We deal with sequential topologies on complete Boolean algebras from the point of view of separation axioms. Our motivation comes from the still open Control Measure Problem of D. Maharam. Maharam asked whether every -complete Boolean algebra that carries a strictly positive continuous submeasure admits a -additive measure. The main result is that a necessary and sufficient condition for B to carry a strictly positive Maharam submeasure is that B is ccc and that the space (B, s) is Hausdroff. We also characterize sequential cardinals. We study sequential topologies on complete Boolean algebras in a more general setting. We review some notations from topology and consider those complete Boolean algebras for which the sequential topology is Fréchet. A necessary and sufficient condition for (B, s) to be a Fréchet space is also one of our important results. We show that for complete ccc Boolean algebra, Hausdorffness of the sequential topology is a strong property : it implies metrizability, and equivalently, the existence of a strictly positive Maharam submeasure. We continue the investigation of order convergence and related topologies on orthomodular lattices (OMLs). Our main results explain why atomic OMLs behave much better than arbitrary ones, not only from the algebraic, but also from the topological point of view: the complete atomic and meet-continuous OMLs are just the order-topological ones, i.e., those complete OMLs which have topological order convergence and form a topological lattice with respect to the order topology; in this situation, the orthocomplementation automatically becomes continuous, being a dual automorphism. Moreover, we shall find that these OMLs are precisely the algebraic (= compactly generated complete) ones, and that it even suffices to postulate continuity in the sense of Scott in order to ensure atomicity. This will be achieved by applying earlier results of the M. Erné to blocks, i.e., to the maximal Boolean subalgebras of the given OML. Among other characterizations, we shall find that a complete OML is order-topological iff it is a totally separated (and, moreover, a totally order-disconnected) topological lattice in its order topology. Our theory essentially extends the compact case, because there are interesting order-topological complete OMLs which fails to be compact. In contrast to this fact, an order-topological complete Boolean algebra is always compact. Several purely algebraic charac-terizations of compact order-topological OMLs will be given. The incomplete case is not yet settled entirely as yet, however, we are able to establish some necessary and sufficient conditions for an OMLs to have a MacNeille completion which is an order-topological atomic OML. Our research work is divided into four chapters with a number of sections and subsections : Chapter I : INTRODUCTION AND PRELIMINARIES It is an introductory chapter. We discuss basic terminologies that are required for smooth understanding of this thesis. This chapter also depicts diagrammatic representation of inclusion relationships among some important subclasses of lattices. This gives us the background needed to begin our exploration “On Problems of von Neumann and Maharam.” Chapter II : CONDITIONS FOR A COMPLETE BOOLEAN ALGEBRA TO CARRY MAHARAM SUBMEASURE We trace out a number of necessary and sufficient conditions for a Boolean -algebra satisfying countable chain condition to carry Maharam submeasure. We remark that weak distributive of Boolean algebra is not necessary in reformulation of Housdorffness and is necessary in reformulation of the G property. Chapter III : TOPOLOGY DETERMINED BY CONVERGENT SEQUENCES IN A COMPLETE BOOLEAN ALGEBRA We investigate the sequential topology s on a complete Boolean algebra B determined by algebraically convergent sequences in B. We have established conditions for (B, s) to a Fréchet space and to be a Hausdroffspace. We show that for complete Boolean algebra satisfying countable chain condition, Hausdorffness of the sequential topology is a strong property. Chapter IV : COMPACT TOPOLOGICAL ORTHOMODULAR LATTICES We show that the following properties are equivalent for any complete orthomodular lattices L: (a) L is order-topological. (b) L is continuous. (a) L is algebraic. (b) L is compactly atomistic. (c) L is a totally order-disconnected topological lattice in the order topology.