"the results of the book are very interesting and profound and can be read successfully without preliminary knowledge. It is written with a great didactical mastery, clearly and precisely It can be recommended not only for specialists on integration theory, but also for a large scale of readers, mainly for postgraduate students".Mathematics Abstracts

The main topics of this book are convergence and topoligization. Integration on a compact interval on the real line is treated with Riemannian sums for various integration bases. General results are specified to a spectrum of integrations, including Lebesgue integration, the Denjoy integration in the restricted sense, the integrations introduced by Pfeffer and by Bongiorno, and many others. Morever, some relations between integration and differentiation are made clear.

The book is self-contained. It is of interest to specialists in the field of real functions, and it can also be read by...

- Contains a thorough treatment of each of the integrals of Riemann, Lebesgue, Henstock-Kurzweil and McShane
- Discusses the weaknesses of the various integrals and presents a comparison of the integrals
- Abundant supply of exercises and examples
- Chapters can be used independently

The relatively new concepts of the Henstock-Kurzweil and McShane integrals based on Riemann type sums are an interesting challenge in the study of integration of Banach space-valued functions. This timely book presents an overview of the concepts developed and results achieved during the past 15 years. The Henstock-Kurzweil and McShane integrals play the central role in the book. Various forms of the integration are introduced and compared from the viewpoint of their generality. Functional analysis is the main tool for presenting the theory of summation gauge integrals.

- Contains a thorough treatment of each of the integrals of Riemann, Lebesgue, Henstock-Kurzweil and McShane
- Discusses the weaknesses of the various integrals and presents a comparison of the integrals
- Abundant supply of exercises and examples
- Chapters can be used independently

The contemporary approach of J. Kurzweil and R. Henstock to the Perron integral is applied to the theory of ordinary differential equations in this book. It focuses mainly on the problems of continuous dependence on parameters for ordinary differential equations. For this purpose, a generalized form of the integral based on integral sums is defined. The theory of generalized differential equations based on this integral is then used, for example, to cover differential equations with impulses or measure differential equations. Solutions of generalized differential equations are found to be...

The Henstock-Kurzweil integral, which is also known as the generalized Riemann integral, arose from a slight modification of the classical Riemann integral more than 50 years ago. This relatively new integral is known to be equivalent to the classical Perron integral; in particular, it includes the powerful Lebesgue integral. This book presents an introduction of the multiple Henstock-Kurzweil integral. Along with the classical results, this book contains some recent developments connected with measures, multiple integration by parts, and multiple Fourier series. The book can be understood...

The book uses classical problems to motivate 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 integration 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 could be used separately in teaching a portion of an introductory real analysis course. There is a sufficient supply of exercises to make this...