In the Introduction to Volume 3 (entitled “Measure algebras”) one can read “…very large parts of the theory, including some of the topics already treated in Volume 2, can be expressed in an appropriately abstract language in which negligible sets have been factored out. This is what the present volume is about. A ‘measure algebra’ is a quotient of an algebra of measurable sets by a null ideal; that is, the elements of the measure algebra are equivalence classes of measurable sets. At the cost of an extra layer of abstraction, we obtain a language which can give concise and elegant expression to a substantial proportion of the ideas of measure theory, and which offers insights almost everywhere in the subject. […] In the structure of this volume I can distinguish seven ‘working’ and two ‘accessory’ chapters. The ‘accessory’ chapters are 31 and 35”.
In the first of ‘accessory chapters’ (entitled “Boolean algebras”) the author presents these results connected with this topic which will be useful later on. The first paragraph introduces the reader to the basic notions and statements. The principal result is Stone’s theorem. The paragraph 312 is very good described in the introduction: “The first part (312A–312K) concerns the translation of the basic concepts of ring theory into the language which I propose to use for Boolean algebras. 312L shows that the order relation on a Boolean algebra defines the algebraic structure, and in 312M–312N I give a fundamental result on the extension of homomorphisms. I end the section with results relating the previous ideas to the Stone representation of a Boolean algebra (312O–312S)”. In the next two paragraphs the notions corresponding with upper (lower) bounds, “sup” and “inf” are considered. The paragraph 315 is devoted to two algebraic constructions: products (simple product) and free products of Boolean algebras.
The main ideas of the results contained in Chapter 32 are best conveyed by the author: “§321 gives the definition of ‘measure algebra’, and relates this idea to its origin as the quotient of a \(\sigma\)-algebra of measurable sets by a \(\sigma\)-ideal of negligible sets, both in its elementary properties and in an appropriate version of the Stone representation. §322 deals with the classification of measure algebras according to the scheme already worked out for measure spaces. §323 discusses the canonical topology and uniformity of a measure algebra. §324 contains results concerning Boolean homomorphisms between measure algebras, with the relationships between topological continuity, order-continuity and preservation of measure. §325 is devoted to the measure algebras of product measures and their abstract characterization. Finally, §§326–327 address the properties of additive functionals on Boolean algebras, generalizing the ideas of Chapter 23”.
The chapter 33 is entitled “Maharam’s theorem”. The first paragraph of this chapter is devoted to the definition and description of homogeneous probability algebras. The main result of this chapter is contained in §332. Let us quote the author: “In this section I present what I call ‘Maharam’s theorem’, that every localizable measure algebra is expressible as a weighted simple product of measure algebras of spaces of the form \(\{ 0,1\}^{\kappa}\) (332B). Among its many consequences is a complete description of the isomorphism classes of localizable measure algebras (332J). This description needs the concepts of ‘cellularity’ of a Boolean algebra (332D) and its refinement, the ’magnitude’ of a measure algebra (332G). I end this section with a discussion of those pairs of measure algebras for which there is a measure-preserving homomorphism from one to the other (332P–332Q), and a general formula for the Maharam type of a localizable measure algebra (332S)”. In the last part of this chapter the author gives some results on the classification of free products of probability algebras.
Paragraph 344 is devoted to the description of three cases in which “the simultaneous realization of countable many homomorphism by a consistent family of functions” can be achieved: Stone spaces, perfect complete countably separated spaces and suitable measures on \(\{ 0,1\}^{I}\). The last results of this part are connected with the statement that two measurable subspaces are isomorphic if and only if they have the same measure. In the §§345 and 346 the author “complements the work of §341 by describing some special properties which can, in appropriate circumstances, be engineered into liftings”.
The Chapter 35 (entitled “Riesz spaces”) is the first of three chapters devoted to an abstract description of the function spaces. The first three paragraphs are connected with the algebraic theory of Riesz spaces (partially ordered linear spaces; general Riesz spaces – a partially ordered linear space which is a lattice and Archimedean and Dedekind complete Riesz spaces). The second part of this chapter is devoted to normed Riesz spaces. In particular the author discusses \(L\)-spaces, \(M\)-spaces, spaces of linear operators and dual Riesz spaces.
The main idea of Chapter 36 (“Function spaces”) is presented in some parts of the introduction. “Here, then, are two of the objectives of this chapter: first, to express the ideas of Chapter 24 in ways making explicit their independence of particular measure spaces, by setting up constructions based exclusively on the measure algebras involved; second, to set out some natural generalizations to other algebras”. In the first paragraph the author presents some problem dealing with the space \(S\) (the Riesz space associated with a boolean ring). The next parts of this chapter are described by the author of the book as follows: “In §362 I seek to relate ideas on additive functional on Boolean algebras from Chapter 23 and §§326–327 to the theory of Riesz space duals in §356. Then, I turn to the systematic discussion of the function spaces of Chapter 24: \(L^{\infty}\) (§363), \(L^{0}\) (§364), \(L^{1}\) (§365) and other \(L^{p}\) (§366), followed by an account of convergence in measure (§367). While all these sections are dominated by the objectives sketched in the paragraphs above, I do include a few major theorems not covered by the ideas of Volume 2, such as the Kelley-Nachbin characterization of the Banach spaces \(L^{\infty}(\mathfrak{A})\) for Dedekind complete \(\mathfrak{A}\) (§363R). In the last two sections of the chapter I turn to the use of \(L^{0}\) spaces in the representation of Archimedean Riesz spaces (§368) and of Banach lattices separated by their order-continuous duals (§369)”.
Chapter 37 is entitled “Linear operators between functions spaces”. The starting point is devoted to an essential property of \(L\)-spaces: if \(U\) and \(V\) are \(L\)-spaces, then every continuous linear operator \(T:U\to V\) is order-bounded. The author gives various generalizations to other \(V\). In §372 Birkhoff Ergodic theorem (also various forms of this theorem) is studied. The purpose of the next paragraph is to discuss operators in the class \({\mathcal T}^{0}\) and two associated classes \({\mathcal T}\) and \({\mathcal T}^{X}\). It is worth noting that the author makes a fuller analysis of the class \({\mathcal T}\), with the characterization of those \(u\), \(v\) such that \(v=Tu\) for some \(T\in {\mathcal T}\). The further part of the chapter one can describe quoting the author: “Using this we can describe ‘rearrangement-invariant’ function spaces and extended Fatou norms (§374). Returning to ideas left on one side in §§364 and 368, I investigate positive linear operators defined on \(L^0\) spaces (§375). In the final section of the chapter (§376), I look at operators which can be defined in terms of kernels on product spaces”.
The next chapter begins with presentations of basic results on authomorphism of general Boolean algebras. §382 contains a general theorem on the expression of an authomorpism as the product of involution. The starting point of §383 are considerations connected with the set of all measure-preserving authomorphisms of \(\mathfrak{A}\) (\(\text{Aut}_{\overline{\mu}}\mathfrak{A}\). The main result contained in this paragraph is following: Let \((\mathfrak{A},\overline{\mu})\) be a localizable measure algebra. Then every measure-preserving authomorphism of \(\mathfrak{A}\) is expressible as the product of at most three measure preserving involutions. In §384 the author presents some results connected with isomorphism between \(\operatorname{Aut}\mathfrak{A}\) and (\(\operatorname{Aut}_{\overline{\mu}} \mathfrak{A}\). Finally let us “partially” quote the author: “I offer two sections on ‘entropy’, the most important numerical invariant enabling us to distinguish some non-conjugate automor-phisms (§§385-386). For Bernoulli shifts on the Lebesgue measure algebra, the isomorphism problem is solved by Ornstein’s theorem; I present a complete proof of this theorem in §§386–387. Finally, in §388, I give Dye’s theorem, describing the full subgroups generated by single automorphisms of measure algebras of countable Maharam type”.
In the last chapter of this book the author presents some statements dealing with the following problem: which other algebras can appear as the underlying Boolean algebra of measure algebra? The content of the chapter the author describes in the following way: “In §391 I discuss algebras which have strictly positive additive real-valued functionals; for such algebras, weak \((\sigma , \infty )\)-distributivity is necessary and sufficient for the existence of a measure; so we are led to look for sufficient conditions to ensure that there is a strictly positive additive functional. A slightly different approach lies through the concept of ‘submeasure’. Submeasures arise naturally in the theories of topological Boolean algebras, topological Riesz spaces and vector measures (see the second half of §393), and on any given algebra there is a strictly positive ‘uniformly exhaustive’ submeasure iff there is a strictly positive additive functional; this is the Kalton-Roberts theorem. It is unknown whether the word ‘uniformly’ can be dropped; this is one of the forms of the Control Measure Problem, which I investigate at length in §393. In §394, I look at a characterization in terms of the special properties which the automorphism group of a measure algebra must have (Kawada’s theorem). §395 complements the previous section by looking briefly at the subgroups of an automorphism group Aut 21 which can appear as groups of measure-preserving automorphisms”.
This volume ends with appendices containing some information relevant to some topics presented in this volume. In this part of the treatise the author gives some facts connected with calculation of cardinalities, cofinalities, Zorn’s lemma, theory of rings, some topics in general topology, uniformities, normed spaces and authomprhisms of groups.
Each section of this treatise ends with “basic exercises”, “further exercises” and “Notes and comments”.