Spaces of measures
[The complete version of the following abridged review is available on demand.] From the preface: ''The theory of spaces of measures may be considered as a chapter of measure and integration theory, if we accept a rather comprehensive definition for the latter theory. In contrast to the other chapters, in which attention is concentrated generally on isolated measures, the properties of sets of measures are of interest for the theory of spaces of measures.'' The book is subdivided into five chapters. The first one contains rather wide collection of definitions and properties of basic tools and objects needed for to present results in the context of spaces of functions. Various spaces of functions used and discussed in the book have many properties which can be established by similar methods. In chapter 2 there is developed an abstract construction which is used later. The third chapter is devoted to spaces of supersummable families. The gist of the book, chapter 4, contains the classical results about spaces of measures. Besides, spaces of additive exhaustive maps and spaces of measures on topological spaces are discussed there. In the final chapter the theory of spaces of measures is used to the study vector lattices and of the M-spaces. The author assumes that the reader has the usual knowledge of a graduate student and is familiar with some special topological objects and integration theory, for the fifth chapter also with the theory of vector lattices and of duality in measure theory. The immense number of well-arranged notions and results makes the monograph very interesting and consistent. On the other hand, the great condensation and abstractness may cause difficulties by reading the book. Contents: Introduction, Logical connections between sections, Notations and terminology, Chapter 1: Topological preliminaries, Chapter 2: Spaces of functions, Chapter 3: Spaces of supersummable families, Chapter 4: Spaces of measures, Chapter 5: Locally convex lattices, References, Index, Notations.
scientific article; zbMATH DE number 3884416