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Measures on topological spaces. (English) Zbl 0910.28009
This survey presents a complete, systematic and very well documented exposition of the integration theory on topological spaces. The central topics are: regularity, transformations and convergence of measures on topological spaces. It contains nine chapters. In Chapter 1, some set-theoretic preliminaries and topological concepts are given. In Chapter 2, various sigma-fields in topological spaces are introduced and the Souslin operation is defined. Chapter 3 deals with several regularity properties of measures, and Baire, Borel, Radon, and perfect measures are treated. Chapter 4 considers the regularity of measures in terms of functionals, and the Riesz-Markov theorems. Radon spaces (and related concepts) and the Skorohod topology are discussed in Chapter 5, while Chapter 6 is dedied to comment several questions about the transformations of measures, such as images of measures, invariant measures of transformations, liftings and conditional measures. Chapter 7 discusses the isomorphisms of measurable spaces and Chapter 8 presents an introduction to the convergence of measures. Here the main definitions and the most important results connected with weak convergence and weak compactness (including Prokhorov spaces), as well as numerous examples, are presented. Finally, Chapter 9 can be regarded as an introduction to the measure theory on topological linear spaces (and mainly on locally convex spaces). Some results are presented with complete proofs, in particular, this concerns a number of new results and examples. Furthermore, a list of 562 references is also given.

MSC:
28C15 Set functions and measures on topological spaces (regularity of measures, etc.)
28-02 Research exposition (monographs, survey articles) pertaining to measure and integration
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