"This book aims at the definition and development of the Henstock-Hurzeil integral and those of the McShane integral in the real line." "The Henstock-Kurzweil integral is of super Lebesgue power while the McShane integral is of Lebesgue power. For bounded functions, however, the Henstock-Kurzweil, the McShane and the Lebesgue integrals are equivalent. Owing to their simple construction and easy access, the Generalized Riemann integrals will surely be familiar to physicists, engineers and applied mathematicians. Each chapter of the book provides a good number of solved problems and counter examples along with selected problems left as exercises."--BOOK JACKET.

The Generalized Riemann Integral is addressed to persons who already have an acquaintance with integrals they wish to extend and to the teachers of generations of students to come. The organization of the work will make it possible for the first group to extract the principal results without struggling through technical details which they may find formidable or extraneous to their purposes. The technical level starts low at the opening of each chapter. Thus, readers may follow each chapter as far as they wish and then skip to the beginning of the next. To readers who do wish to see all the details of the arguments, they are given. The generalized Riemann integral can be used to bring the full power of the integral within the reach of many who, up to now, haven't gotten a glimpse of such results as monotone and dominated convergence theorems. As its name hints, the generalized Riemann integral is defined in terms of Riemann sums. The path from the definition to theorems exhibiting the full power of the integral is direct and short.

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.

This is an elementary, self-contained presentation of the integration processes developed by Lebesgue, Denjoy, Perron, and Henstock. An excellent text for graduate students with a background in real analysis.

The derivative and the integral are the fundamental notions of calculus. Though there is essentially only one derivative, there is a variety of integrals, developed over the years for a variety of purposes, and this book describes them. No other single source treats all of the integrals of Cauchy, Riemann, Riemann-Stieltjes, Lebesgue, Lebesgue-Steiltjes, Henstock-Kurzweil, Weiner, and Feynman. The basic properties of each are proved, their similarities and differences are pointed out, and the reasons for their existence and their uses are given, with plentiful historical information. The audience for the book is advanced undergraduate mathematics students, graduate students, and faculty members, of which even the most experienced are unlikely to be aware of all of the integrals in the Garden of Integrals. Professor Burk's clear and well-motivated exposition makes this book a joy to read. There is no other book like it.

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.

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 with a prerequisite of advanced calculus.

The Lebesgue integral is now standard for both applications and advanced mathematics. This books starts with a review of the familiar calculus integral and then constructs the Lebesgue integral from the ground up using the same ideas. A Primer of Lebesgue Integration has been used successfully both in the classroom and for individual study. Bear presents a clear and simple introduction for those intent on further study in higher mathematics. Additionally, this book serves as a refresher providing new insight for those in the field. The author writes with an engaging, commonsense style that appeals to readers at all levels.

Every good mathematical book stands like a tree with its roots in the past and its branches stretching out towards the future. Whether the fruits of this tree are desirable and whether the branches will be quarried for mathematical wood to build further edifices, I will leave to the judgment of history. The roots of this book take nourishment from the concept of definite integration of continuous functions, where Riemann's method is the high water mark of the simpler theory.

This book presents the Henstock/Kurzweil integral and the McShane integral. These two integrals are obtained by changing slightly the definition of the Riemann integral. These variations lead to integrals which are much more powerful than the Riemann integral. The Henstock/Kurzweil integral is an unconditional integral for which the fundamental theorem of calculus holds in full generality, while the McShane integral is equivalent to the Lebesgue integral in Euclidean spaces. A basic knowledge of introductory real analysis is required of the reader, who should be familiar with the fundamental properties of the real numbers, convergence, series, differentiation, continuity, etc. Contents: Introduction to the Gauge or Henstock-Kurzweil Integral; Basic Properties of the Gauge Integral; Henstock''s Lemma and Improper Integrals; The Gauge Integral over Unbounded Intervals; Convergence Theorems; Integration over More General Sets: Lebesgue Measure; The Space of Gauge Integrable Functions; Multiple Integrals and Fubini''s Theorem; The McShane Integral; McShane Integrability is Equivalent to Absolute Henstock-Kurzweil Integrability. Readership: Upper level undergraduates and mathematicians interested in gauge integrals.