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.
Rezensionen / Stimmen
...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 *
Sprache
Verlagsort
Zielgruppe
Für höhere Schule und Studium
Illustrationen
Maße
Höhe: 241 mm
Breite: 161 mm
Dicke: 22 mm
Gewicht
ISBN-13
978-0-19-850124-4 (9780198501244)
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Schweitzer Klassifikation
Autor*in
Reader in MathematicsReader in Mathematics, University of Oxford
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
