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Zusatztext 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. Klappentext 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 examples and exercises. Intended as a first course in integration theory for students familiar with real analysis! the book begins with a simplified Lebesgue integral! which is then developed to provide an entry point for important results in the field. The final chapters present selected applications! mostly drawn from Fourier analysis. The emphasis throughout is on integrable functions rather than on measures. Designed as an undergraduate or graduate textbook! it is a companion volume to the author's Introduction to Complex Analysis and is aimed at both pure and applied mathematicians. Zusammenfassung This account of integration theory gives a practical guide to the Lebesgue integral with examples. Aimed at both pure and applied mathematicians, it begins with a simplified integral intended for a first course in integration, serving as introduction to Lebesgue integral proper. Last chapters present selected applications to Fourier analysis. Inhaltsverzeichnis 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 ...
