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Cambridge University Press, Published 1993. Hardcover, 302 pp.; 24 cm; First Edition. In Very Good condition with a Good dust jacket. Dark blue cloth boards, silver lettering on spine, mild shelf and edge wear with some color loss along edges and spine hinges. Binding tight. Pages clean and unmarked, a few small spots to rear endpaper. Light blue dust jacket, several nicks and closed tears of half an inch or less, creasing along edges and at flap folds, some soiling, and a semi-circular stain to paper over top edge of front cover near corner. Otherwise unmarked. Now in an archival quality (removable) Brodart cover. Number 109 in Cambridge Tracts in Mathematics, a series of single-thread monographs intended as definitive treatments. [From front flap] This book presents a detailed and mostly elementary exposition of the generalized Riemann integrals discovered by Henstock, Kurzweil, and McShane. Along with the classical results, it contains some recent developments connected with lipeomorphic change of variables, higher dimensional multipliers, and the divergence theorem for discontinuously differentiable vector fields. Defining the Lebesgue integral in Euclidean spaces from McShane's point of view has a clear pedagogical advantage, since the initial stages of development are both conceptually and technically simpler. The McShane integral evolves directly from the basic ideas about integration taught in elementary calculus. The difficult transition from subdividing the domain to subdividing the range, intrinsic to the Lebesgue definition, is completely bypassed. The unintuitive Caratheodory concept of measurability is also made more palatable by means of locally fine partitions. The mathematical significance of the generalized Riemann integrals emerges when the Henstock-Kurzweil approach is used to define the Denjoy-Perron integral. While there is little similarity between the classical definitions of Lebesgue, Denjoy, and Perron, the Lebesgue and Denjoy-Perron integrals are naturally connected via the McShane and Henstock-Kurzweil definitions. This is used to obtain a coordinate free multidimensional integral which provides an unrestricted Gauss-Green theorem for vector fields with large sets of singularities. Although written as a monograph, the book can be used as a graduate text, and certain portions of it can be presented even to advanced undergraduate students with a working knowledge of limits, continuity, and differentiation on the real line. Contents: Preface; Acknowledgments; One-dimensional integration: Preliminaries, The McShane integral, Measure and measurability, Integrable functions, Descriptive definition, The Henstock-Kurzweil integral; Multidimensional integration: Preliminaries, The McShane integral, Descriptive definition, Change of variables, The gage integral, The F-integral, Recent developments; Bibliography; List of symbols; Index.
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