This text is intended as a treatise for a rigorous course introducing the elements of integration theory on the real line. All of the important features of the Riemann integral, the Lebesgue integral, and the Henstock-Kurzweil integral are covered.
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This book is licensed under a Creative Commons License (CC BY-NC-SA). You can download the ebook Theory of the Integral for free.
- Title
- Theory of the Integral
- Author(s)
- Brian S Thomson
- Published
- 2013-02-10
- Edition
- 1
- Format
- eBook (pdf, epub, mobi)
- Pages
- 422
- Language
- English
- ISBN-10
- 1467924393
- ISBN-13
- 9781467924399
- License
- CC BY-NC-SA
- Book Homepage
- Free eBook, Errata, Code, Solutions, etc.
Preface Table of Contents By way of an introduction The classical Newton integral Bounded integrable functions and Lipschitz functions Absolutely integrable functions and bounded variation The Newton integral is a nonabsolute integral Continuity and integrability Upper functions Continuous functions are Newton integrable Proof of Lemma 1.5 Riemann sums Mean-value theorem and Riemann sums Uniform Approximation by Riemann sums Cauchy's theorem Robbins's theorem Proof of Theorem 1.9 Characterization of Newton's integral Proof of Theorem 1.10 How to generalize the integral? What sets to ignore? Exceptional sets Sets of measure zero Proof of Lemma 1.12 Zero variation Generalized Newton integral Exercises Newton integral: controlled version Proof of Theorem 1.16 Continuous linear functionals Which Newton variant should we teach Constructive aspects: the regulated integral Step functions and regulated functions Riemann's integral Integrability criteria Proof of Theorem 1.21 Volterra's Example Integral of Henstock and Kurzweil A Cauchy criterion The Henstock-Saks Lemma Proof of Theorem 1.24 The Henstock-Kurzweil integral includes all Newton integrals Integral of Lebesgue The Cauchy property Lebesgue differentiation theorem Bounded variation The Dini derivatives Two easy lemmas Proof of the Lebesgue differentiation theorem Removing the continuity hypothesis Infinite integrals Where are we? Appendix: Constructive vs. descriptive Covering Theorems Covering Relations Partitions and subpartitions Covering relations Prunings Full covers Fine covers Uniformly full covers Covering arguments Cousin covering lemma A simple covering argument Decomposition of full covers Riemann sums Sets of Lebesgue measure zero Lebesgue measure of open sets Sets of Lebesgue measure zero Sequences of Lebesgue measure zero sets Almost everywhere a.e. language Full null sets Fine null sets The Mini-Vitali Covering Theorem Covering lemmas for families of compact intervals Proof of the Mini-Vitali covering theorem Functions having zero variation Zero variation and zero derivatives Generalization of the zero derivative/variation Zero variation and mapping properties Absolutely continuous functions Absolute continuity in the sense of Vitali Proof of Lemma 2.30 Absolute continuity in the variational sense Absolute continuity and derivatives An application to the Henstock-Kurzweil integral Proof of Theorem 2.35 Lebesgue differentiation theorem (2nd proof) Upper and lower derivates Geometrical lemmas Proof of the Lebesgue differentiation theorem Fubini differentiation theorem An application to the Riemann integral Riesz's problem Other variants The Integral Upper and lower integrals The integral and integrable functions HK criterion Cauchy criterion McShane's criterion Elementary properties of the integral Integration and order Integration of linear combinations Integrability on subintervals Additivity A simple change of variables Integration by parts Derivative of the integral Null functions Monotone convergence theorem Summing inside the integral Two convergence lemmas Equi-integrability Lebesgue's Integral The Lebesgue integral Lebesgue measure Basic property of Lebesgue measure Vitali covering theorem Classical version of Vitali's theorem Proof that = * = * . Density theorem Approximate Cousin lemma Additivity Measurable sets Definition of measurable sets Properties of measurable sets Increasing sequences of sets Existence of nonmeasurable sets Measurable functions Measurable functions are almost bounded Continuous functions are measurable Derivatives are measurable Integrable functions are measurable Simple functions Measurable functions are almost simple Limits of measurable functions Construction of the integral Characteristic functions of measurable sets Characterizations of measurable sets Integral of simple functions Integral of bounded measurable functions Integral of nonnegative measurable functions Fatou's Lemma Dominated convergence theorem Derivatives Derivative of the integral Lebesgue points and points of approximate continuity Derivatives of functions of bounded variation Characterization of the Lebesgue integral McShane's Criterion Nonabsolutely integrable functions Convergence of sequences of functions Review of elementary theory Modes of convergence Comparison of modes of convergence on [a,b] A sliding sequence of functions Riesz's Theorem Egorov's Theorem Dominated convergence on an interval Lusin's theorem Littlewood's three principles Denjoy-Stepanoff theorem Absolute continuity of the integral Convergence and equi-integrability Equi-integrability A stronger convergence theorem McShane equi-integrability condition Young-Daniell-Riesz Program Recall the program for the regulated integral Riesz sequences Convergence of Riesz sequences of step functions Representing semicontinuous functions by Riesz sequences Characterization of the Lebesgue integral Vitali-Carathéodory property Exercises Characterizations of the indefinite integral Indefinite integral of nonnegative, integrable functions Indefinite integral of Lebesgue integrable functions Indefinite integral of nonabsolutely integrable functions Proofs Extending Lebesgue's integral The Lebesgue integral as a set function The abstract Lebesgue integral Assumptions of the general theory Defining the integral Properties of the integral Stieltjes Integrals Stieltjes integrals Definition of the Stieltjes integral Definition of the Riemann-Stieltjes integral Henstock's zero variation criterion Regulated functions Approximate additivity of naturally regulated functions Variation expressed as an integral Representation theorems Jordan decomposition Jordan decomposition theorem: differentiation Representation by saltus functions Representation by singular functions Reducing a Stieltjes integral to an ordinary integral Properties of the indefinite integral Existence of the integral from derivative statements Existence of the integral for continuous functions Integration by parts Lebesgue-Stieltjes measure Mutually singular functions Singular functions Length of curves Formula for the length of curves Change of variables Easy change of variables Another easy change of variables A general change of variables Change of variables for Lipschitz functions Theorem of Kestelman, Preiss, and Uher Nonabsolutely Integrable Functions Variational Measures Full and fine variational measures Finite variation and -finite variation The Vitali property Kolmogorov equivalence Variation of continuous, increasing functions Variation and image measure Variational classifications of real functions Derivates and variation Ordinary derivates and variation Dini derivatives and variation Lipschitz numbers Six growth lemmas Continuous functions with -finite variation Variation on compact sets -absolutely continuous functions Vitali property and differentiability The Vitali property and variation Monotonic functions Functions of bounded variation Functions of -finite variation Characterization of the Vitali property Characterization of -absolute continuity Mapping properties Lusin's conditions Banach-Zarecki Theorem Local Lebesgue integrability conditions Continuity of upper and lower integrals A characterization of the integral Denjoy's program Integration method Cauchy extension Harnack extension Transfinite sequence of extensions of the integral The totalization process The Perron-Bauer program Major and minor functions Major and minor functions applied to other integrals The Perron ``integral'' Hake-Alexandroff-Looman theorem Marcinkiewicz theorem Integral of Dini derivatives Motivation Quasi-Cousin covering lemma Estimates of integrals from derivates Estimates of integrals from Dini derivatives Appendix: Baire category theorem Meager sets Portions Baire-Osgood Theorem Language of meager/residual subsets Integration in Rn Some background Intervals and covering relations Measure and integral Lebesgue measure in Rn The fundamental lemma Measurable sets and measurable functions Measurable functions Notation General measure theory Iterated integrals Formulation of the iterated integral property Fubini's theorem Expression as a Stieltjes integral ANSWERS Answers to problems Bibliography Index
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