Theories Of Integration

This book presents a historical development of the integration theories of Riemann, Lebesgue, Henstock-Kurzweil, and McShane, showing how new theories of integration were developed to solve problems that earlier theories could not handle. It develops the basic properties of each integral in detail and provides comparisons of the different integrals. The chapters covering each integral are essentially independent and can be used separately in teaching a portion of an introductory course on real analysis. There is a sufficient supply of exercises to make the book useful as a textbook.

The theory of integration is one of the twin pillars on which analysis is built. The first version of integration that students see is the Riemann integral. Later, graduate students learn that the Lebesgue integral is ``better'' because it removes some restrictions on the integrands and the domains over which we integrate. However, there are still drawbacks to Lebesgue integration, for instance, dealing with the Fundamental Theorem of Calculus, or with ``improper'' integrals. This book is an introduction to a relatively new theory of the integral (called the ``generalized Riemann integral'' or the ``Henstock-Kurzweil integral'') that corrects the defects in the classical Riemann theory and both simplifies and extends the Lebesgue theory of integration. Although this integral includes that of Lebesgue, its definition is very close to the Riemann integral that is familiar to students from calculus. One virtue of the new approach is that no measure theory and virtually no topology is required. Indeed, the book includes a study of measure theory as an application of the integral. Part 1 fully develops the theory of the integral of functions defined on a compact interval. This restriction on the domain is not necessary, but it is the case of most interest and does not exhibit some of the technical problems that can impede the reader's understanding. Part 2 shows how this theory extends to functions defined on the whole real line. The theory of Lebesgue measure from the integral is then developed, and the author makes a connection with some of the traditional approaches to the Lebesgue integral. Thus, readers are given full exposure to the main classical results. The text is suitable for a first-year graduate course, although much of it can be readily mastered by advanced undergraduate students. Included are many examples and a very rich collection of exercises. There are partial solutions to approximately one-third of the exercises. A complete solutions manual is available separately.

With coverage of both traditional and critical theories and approaches to European integration and their application, this is the most comprehensive textbook on European integration theory and an essential guide for all students and scholars interested in the subject. Throughout the text, a team of leading international scholars demonstrate the current relevance of integration theory as they apply these approaches to real-world developments and crises in the contemporary European Union.

A European Security Architecture after the Cold War provides a critical account of the re-projection and redefinition of Western values and security institutions in the post-Coldwar era. This transformation is explored in three stages. The first stage covers the period 1990-91 and explains the preservation of a `western security community' inherited from the Cold War, through a process of institutional reconstruction largely carried out on paper. The second stage from 1991 to 1992 sees the incorporation of a `purpose' for these institutions as a framework for the implementation of collective security. The third stage explores the emerging questions of legitimacy surrounding the new tasks of these institutions as they become embroiled in the war in the former Yugoslavia. The precedents of legitimate intervention in upholding democracy, free markets and human rights in the post-coldwar era are examined from the perspectives of international law and Gramscian derived concepts of legitimacy, focusing on the acceptance of military power by civil society, and how intervention in these terms becomes a 'cultural practice'.

This text offers a multidisciplinary overview of theories of, and academic approaches to, European integration. The authors include four political scientists, an economist, a historian and a legal scholar. They examine critically the theories of European integration, as well as related theoretical and empirical works in political science, sociology and economics.

The authors engage a dialogue between European integration theories and gender studies. The contributions illustrate where and how gender scholarship has made creative use of integration theories and thus contributes to a vivid theoretical debate. The chapters are designed to make gender scholarship more visible to integration theory and, in this way stimulates the broader theoretical debates. Investigating the whole range of integration theory with a gender lens, the authors illustrate if and how gender scholarship has made or can make creative use of integration theories.

In this first book-length treatment of integration theory for many years, Ben Rosamond provides an accessible and stimulating critical introduction to the full range of classical and contemporary perspectives. The book explains the centrality of theoretical work to the study of integration and the EU and carefully locates different theories within their wider intellectual and 'real world' contexts. This thoroughly researched book engages with the key debates to have arisen from theoretical deliberations about European integration. It develops its own distinctive contribution, emphasising the importance of 'sociology of knowledge' questions when evaluating integration theory and stressing the continued significance of international theory to the study of the EU.