Advanced Integration Theory / Edition 1

by Corneliu Constantinescu, Alexia Sontag, Wolfgang Filter, Karl Weber

Overview

Since about 1915 integration theory has consisted of two separate branches: the abstract theory required by probabilists and the theory, preferred by analysts, that combines integration and topology. As long as the underlying topological space is reasonably nice (e.g., locally compact with countable basis) the abstract theory and the topological theory yield the same results, but for more compli­ cated spaces the topological theory gives stronger results than those provided by the abstract theory. The possibility of resolving this split fascinated us, and it was one of the reasons for writing this book. The unification of the abstract theory and the topological theory is achieved by using new definitions in the abstract theory. The integral in this book is de­ fined in such a way that it coincides in the case of Radon measures on Hausdorff spaces with the usual definition in the literature. As a consequence, our integral can differ in the classical case. Our integral, however, is more inclusive. It was defined in the book "C. Constantinescu and K. Weber (in collaboration with A.

Product Details

ISBN-13:	9780792352341
Publisher:	Springer Netherlands
Publication date:	10/31/1998
Series:	Mathematics and Its Applications , #454
Edition description:	1998
Pages:	876
Product dimensions:	6.10(w) x 9.25(h) x 0.07(d)

Table of Contents

Preface. Introduction. Suggestions to the Reader. 0. Preliminaries. 1. Vector Lattices. 2. Elementary Integration Theory. 3. Lp-Spaces. 4. Real Measures. 5. The Radon-Nikodym Theorem Duality. 6. The Classical Theory of Real Functions. Historical Remarks. Name Index. Subject Index. Symbol Index.

Editorial Reviews

This book gives a general definition of the (abstract) integral using the Daniell method. A consequence of this approach is that integration theory on Hausdorff topological spaces appears to be a special case of abstract integration theory. The theory of vector lattices is presented, since it is a tool for the development of the abstract theory. Its consequent application not only yields new insight into integration theory, but also simplifies many proofs. For example, the space of real-valued measures on a d-ring turns out to be an order complete vector lattice, which permits a coherent development of the theory and the elegant derivation of many classical and new results. Exercises accompany each section. For graduate students and researchers. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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