Introduction to Integration

by Priestley, H. A.

• ISBN13: 9780198501237
• ISBN10: 0198501234
• Format: Paperback
• Copyright: 1997-12-04
• Publisher: Clarendon 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

Setting the scene

Preliminaries

Intervals and step functions

Integrals of step functions

Continuous functions on compact intervals

Techniques of Integration I

Approximations

Uniform convergence and power series

Building foundations

Null sets

Linc functions

The space L of integrable functions

Non-integrable functions

Convergence Theorems: MCT and DCT

Recognizing integrable functions I

Techniques of integration II

Sums and integrals

Recognizing integrable functions II

The Continuous DCT

Differentiation of integrals

Measurable functions

Measurable sets

The character of integrable functions

Integration VS. differentiation

Integrable functions of Rk

Fubini's Theorem and Tonelli's Theorem

Transformations of Rk

The spaces L1, L2 and Lp

Fourier series: pointwise convergence

Fourier series: convergence re-assessed

L2-spaces: orthogonal sequences

L2-spaces as Hilbert spaces

The Fourier transform

Integration in probability theory

Appendix I

Appendix II

Bibliography

Notation index

Subject index

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