Introduction to Integration

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Clarendon Press, 1997 - Mathematics - 306 pages
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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 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.
 

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Contents

Setting the scene
1
Preliminaries
8
Intervals and step functions
21
Integrals of step functions
29
Continuous functions on compact intervals
34
Techniques of integration I
44
Approximations
56
Uniform convergence and power series
67
Measurable functions
160
Measurable sets
166
The character of integrable functions
172
Integration vs differentiation
177
Integrable functions on Rfc
189
Fubinis Theorem and Tonellis Theorem
190
Transformations of Rk
209
The spaces L1 L2 and L?
210

Building foundations
78
Null sets
87
Linc functions
93
The class L of integrable functions
102
Nonintegrable functions
110
MCT and DCT
117
Recognizing integrable functions I
125
Techniques of integration II
132
Sums and integrals
137
Recognizing integrable functions II
143
The Continuous DCT
148
Differentiation of integrals
152
pointwise convergence
221
convergence reassessed
236
orthogonal sequences
247
L2spaces as Hilbert spaces
258
Fourier transforms
264
Integration in probability theory
279
historical remarks
287
reference
291
Bibliography
295
Notation index
297
Subject index
299
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About the author (1997)

Hilary Priestley is at University of Oxford.

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