Introduction to Integration

by Priestley, H. A.

• ISBN13: 9780198501244
• ISBN10: 0198501242
• Format: Hardcover
• Copyright: 1997-12-04
• Publisher: Oxford University Press

Summary

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.

Table of Contents

Notation	xi
1. Setting the scene	1	(7)
2. Preliminaries	8	(13)
3. Intervals and step functions	21	(8)
4. Integrals of step functions	29	(5)
5. Continuous functions on compact intervals	34	(10)
6. Techniques of integration I	44	(12)
7. Approximations	56	(11)
8. Uniform convergence and power series	67	(11)
9. Building foundations	78	(9)
10. Null sets	87	(6)
11. L(inc) functions	93	(9)
12. The class L of integrable functions	102	(8)
13. Non-integrable functions	110	(7)
14. Convergence Theorems: MCT and DCT	117	(8)
15. Recognizing integrable functions I	125	(7)
16. Techniques of integration II	132	(5)
17. Sums and integrals	137	(6)
18. Recognizing integrable functions II	143	(5)
19. The Continuous DCT	148	(4)
20. Differentiation of integrals	152	(8)
21. Measurable functions	160	(6)
22. Measurable sets	166	(6)
23. The character of integrable functions	172	(5)
24. Integration vs. differentiation	177	(7)
25. Integrable functions on R(k)	184	(6)
26. Fubini's Theorem and Tonelli's Theorem	190	(11)
27. Transformations of R(k)	201	(9)
28. The spaces L(1), L(2), and L(p)	210	(11)
29. Fourier series: pointwise convergence	221	(15)
30. Fourier Series: convergence reassessed	236	(11)
31. L(2)-spaces: orthogonal sequences	247	(11)
32. L(2)-spaces as Hilbert spaces	258	(6)
33. Fourier transforms	264	(15)
34. Integration in probability theory	279	(8)
Appendix I: historical remarks	287	(4)
Appendix II: reference	291	(4)
Bibliography	295	(2)
Notation index	297	(2)
Subject index	299