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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 traditional Riemann 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 functions rather 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.
Readership: This book is aimed at mathematics undergraduates, specifically those doing a course on the Lebesgue integral or the theory of integration. Later chapters would be a supplementary reference for courses on Fourier analysis and functional analysis.
H. A. Priestley, Reader in Mathematics, University of Oxford
"Delightful book. Those who know Hilary Priestley will recognise at once the impish sense of fun which permeates this book (even down to the selection of notation): she has a real gift for the memorable phrase and the agonies oand ecstasies of teaching 25 years worth of Oxford undergraduates are etched in the motivational and orientational remarks, helpful reiterations of key points, local stock-taking' susummaries and tight internal sign-posting. By its very nature integration theory cannot be made easy, but Professor Priestley will rapidly earn the gratitude of a new generation of students for making it as pleasantly palatable as one could wish for."
"This is a very readable and well-planned book, most suitable for all mathematics graduates. The emphasis is on practice with many applications in the later chapters."
1: Setting the scene 2: Preliminaries 3: Intervals and step functions 4: Integrals of step functions 5: Continuous functions on compact intervals 6: Techniques of Integration I 7: Approximations 8: Uniform convergence and power series 9: Building foundations 10: Null sets 11: Linc functions 12: The space L of integrable functions 13: Non-integrable functions 14: Convergence Theorems: MCT and DCT 15: Recognizing integrable functions I 16: Techniques of integration II 17: Sums and integrals 18: Recognizing integrable functions II 19: The Continuous DCT 20: Differentiation of integrals 21: Measurable functions 22: Measurable sets 23: The character of integrable functions 24: Integration VS. differentiation 25: Integrable functions of Rk 26: Fubini's Theorem and Tonelli's Theorem 27: Transformations of Rk 28: The spaces L1, L2 and Lp 29: Fourier series: pointwise convergence 30: Fourier series: convergence re-assessed 31: L2-spaces: orthogonal sequences 32: L2-spaces as Hilbert spaces 33: The Fourier transform 34: Integration in probability theory Appendix I Appendix II Bibliography Notation index Subject index
