# Introduction to Integration

Clarendon Press, 1997 - Mathematics - 306 pages
Introduction to integration provides a unified account of integration theory, giving a practical guide to the Lebesgue integral and its uses, with a wealth of illustrative examples and exercises. The book begins with a simplified Lebesgue-style integral (in lieu of the more traditionalRiemann integral), intended for a first course in integration. This suffices for elementary applications, and serves as an introduction to the core of the book. The final chapters present selected applications, mostly drawn from Fourier analysis. The emphasis throughout is on integrable functionsrather than on measure. The book is designed primarily as an undergraduate or introductory graduate textbook. It is similar in style and level to Priestley's Introduction to complex analysis, for which it provides a companion volume, and is aimed at both pure and applied mathematicians.Prerequisites are the rudiments of integral calculus and a first course in real analysis.

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### Contents

 Setting the scene 1 Preliminaries 8 Intervals and step functions 21 Integrals of step functions 29 Continuous functions on compact intervals 34 Techniques of integration I 44 Approximations 56 Uniform convergence and power series 67
 Measurable functions 160 Measurable sets 166 The character of integrable functions 172 Integration vs differentiation 177 Integrable functions on Rfc 189 Fubinis Theorem and Tonellis Theorem 190 Transformations of Rk 209 The spaces L1 L2 and L? 210

 Building foundations 78 Null sets 87 Linc functions 93 The class L of integrable functions 102 Nonintegrable functions 110 MCT and DCT 117 Recognizing integrable functions I 125 Techniques of integration II 132 Sums and integrals 137 Recognizing integrable functions II 143 The Continuous DCT 148 Differentiation of integrals 152
 pointwise convergence 221 convergence reassessed 236 orthogonal sequences 247 L2spaces as Hilbert spaces 258 Fourier transforms 264 Integration in probability theory 279 historical remarks 287 reference 291 Bibliography 295 Notation index 297 Subject index 299 Copyright

### About the author (1997)

Hilary Priestley is at University of Oxford.