Concise introduction to the theory of integration,

'This is a very attractive textbook... Unusual in many respects, [it] fully achieves its goal, a one-semester, concise but logically complete treatment of abstract integration. It is remarkable that [the author] has accomplished so much in so short a compass.' -- Mathematical Reviews (on the first edition) 'A number of valuable applications and a good collection of problems... A very interesting, well-informed book which draws on recent approaches not found in several commonly used texts.' -- Zbl. fuer Mathematik (from a review of the second edition) 'The author succeeded in choosing the right level of generality and showed how a good combination of a measure and integration course and advanced calculus can be done. Strongly recommended to students as well as to teachers.' -- EMS Newsletter (from a review of the second edition) Designed for the full-time analyst, physicists, engineer, or economist, this book attempts to provide its readers with most of the measure theory they will ever need. Given the choice, the author has consistently opted to develop the concrete rather than the abstract aspects of topics treated. The major new feature of this third edition is the inclusion of a new chapter in which the author introduces the Fourier transform. In that Hermite functions play a central role in his treatment of Parseval's identity and the inversion formula, Stroock's approach bears greater resemblance to that adopted by Norbert Wiener than that used in most modern introductory texts. A second feature is that solutions to all problems are provided. As a self-contained text, this book is excellent for both self-study and the classroom. TABLE OF CONTENTS Preface to Third Edition Preface to Second Edition Preface to First Edition Chapter I: The Classical Theory 1.1 Riemann Integration 1.2 Riemann-Stieltjes Integration Chapter II: Lebesgue Measure 2.0 The Idea 2.1 Existence 2.2 Euclidean Invariance Chapter III: Lebesgue Integration 3.1 Measure Spaces 3.2 Construction of Integrals 3.3 Convergence of Integrals 3.4 Lebesgue's Differentiation Theorem Chapter IV: Products of Measures 4.1 Fubini's Theorem 4.2 Steiner Symmetrization and the Isodiametric Inequality Chapter V: Changes of Variable 5.0 Introduction 5.1 Lebesgue vs. Riemann Integrals 5.2 Polar Coordinates 5.3 Jacobi's Transformation and Surface Measure 5.4 The Divergence Theorem Chapter VI: Some Basic Inequalities 6.1 Jensen, Minkowski, and Hoelder 6.2 The Lebesgue Spaces 6.3 Convolution and Approximate Identities Chapter VII: Elements of Fourier Analysis 7.1 Hilbert Space 7.2 Fourier Series 7.3 The Fourier Transform, L 1 -theory 7.4 Hermite Functions 7.5 The Fourier Transform, L 2 -theory Chapter VIII: A Little Abstract Theory 8.1 An Existence Theorem 8.2 The Radon-Nikodym Theorem Solution Manual Notation Index