Advanced Integration Theory

✍ Scribed by Corneliu Constantinescu, Wolfgang Filter, Karl Weber, Alexia Sontag (auth.)

• Publisher: Springer Netherlands
• Year: 1998
• Tongue: English
• Leaves: 876
• Series: Mathematics and Its Applications 454
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-x
Introduction....Pages 1-4
Suggestions to the Reader....Pages 5-6
Preliminaries....Pages 7-20
Vector Lattices....Pages 21-278
Elementary Integration Theory....Pages 279-445
L p -Spaces....Pages 447-548
Real Measures....Pages 549-636
The Radon-Nikodym Theorem. Duality....Pages 637-679
The Classical Theory of Real Functions....Pages 681-804
Back Matter....Pages 805-870

✦ Subjects

Measure and Integration; Order, Lattices, Ordered Algebraic Structures; Real Functions; Probability Theory and Stochastic Processes; Integral Transforms, Operational Calculus