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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
"...my appreciation? It is always a touchy discussion when debating about the use of the Riemann (or an analogous to) or the Lebesgue approach for a beginner's course on integration theory. My point of view is quite close to the one of the author so there will be no arguing here. To the others, let me just say the following: take the book in your hands, read (part of) it and you will see that it is reasonable and advisable to present a rigorous introduction of the Lebesgue integration theory to beginners." - Belgian Mathematical Society
"Priestley takes a great deal of care to motivate you to grasp the concepts and introduces plenty of examples." - New Scientist, 3 October 1998
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
