Hardcover ISBN:  9780821829745 
Product Code:  GSM/45 
List Price:  $77.00 
MAA Member Price:  $69.30 
AMS Member Price:  $61.60 
Electronic ISBN:  9781470420963 
Product Code:  GSM/45.E 
List Price:  $72.00 
MAA Member Price:  $64.80 
AMS Member Price:  $57.60 

Book DetailsGraduate Studies in MathematicsVolume: 45; 2002; 424 ppMSC: Primary 28; Secondary 26;
Integration is one of the two cornerstones of analysis. Since the fundamental work of Lebesgue, integration has been interpreted in terms of measure theory. This introductory text starts with the historical development of the notion of the integral and a review of the Riemann integral. From here, the reader is naturally led to the consideration of the Lebesgue integral, where abstract integration is developed via measure theory. The important basic topics are all covered: the Fundamental Theorem of Calculus, Fubini's Theorem, \(L_p\) spaces, the RadonNikodym Theorem, change of variables formulas, and so on.
The book is written in an informal style to make the subject matter easily accessible. Concepts are developed with the help of motivating examples, probing questions, and many exercises. It would be suitable as a textbook for an introductory course on the topic or for selfstudy.
For this edition, more exercises and four appendices have been added.The AMS maintains exclusive distribution rights for this edition in North America and nonexclusive distribution rights worldwide, excluding India, Pakistan, Bangladesh, Nepal, Bhutan, Sikkim, and Sri Lanka.
ReadershipGraduate students and research mathematicians interested in mathematical analysis.

Table of Contents

Chapters

Prologue. The length function

Chapter 1. Riemann integration

Chapter 2. Recipes for extending the Riemann integral

Chapter 3. General extension theory

Chapter 4. The Lebesgue measure on $\mathbb {R}$ and its properties

Chapter 5. Integration

Chapter 6. Fundamental theorem of calculus for the Lebesgue integral

Chapter 7. Measure and integration on product spaces

Chapter 8. Modes of convergence and $L_p$spaces

Chapter 9. The RadonNikodym theorem and its applications

Chapter 10. Signed measures and complex measures

Appendix A. Extended real numbers

Appendix B. Axiom of choice

Appendix C. Continuum hypotheses

Appendix D. Urysohn’s lemma

Appendix E. Singular value decomposition of a matrix

Appendix F. Functions of bounded variation

Appendix G. Differentiable transformations


Reviews

From reviews for the first edition: Distinctive features include: 1) An unusually extensive treatment of the historical developments leading up to the Lebesgue integral … 2) Presentation of the standard extension of an abstract measure on an algebra to a sigma algebra prior to the final stage of development of Lebesgue measure. 3) Extensive treatment of change of variables theorems for functions of one and several variables … the conversational tone and helpful insights make this a useful introduction to the topic … The material is presented with generous details and helpful examples at a level suitable for an introductory course or for selfstudy.
Zentralblatt MATH 
A special feature [of the book] is the extensive historical and motivational discussion … At every step, whenever a new concept is introduced, the author takes pains to explain how the concept can be seen to arise naturally … The book attempts to be comprehensive and largely succeeds … The text can be used for either a onesemester or a oneyear course at M.Sc. level … The book is clearly a labor of love. The exuberance of detail, the wealth of examples and the evident delight in discussing variations and counter examples, all attest to that … All in all, the book is highly recommended to serious and demanding students.
Resonance — journal of science education


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Integration is one of the two cornerstones of analysis. Since the fundamental work of Lebesgue, integration has been interpreted in terms of measure theory. This introductory text starts with the historical development of the notion of the integral and a review of the Riemann integral. From here, the reader is naturally led to the consideration of the Lebesgue integral, where abstract integration is developed via measure theory. The important basic topics are all covered: the Fundamental Theorem of Calculus, Fubini's Theorem, \(L_p\) spaces, the RadonNikodym Theorem, change of variables formulas, and so on.
The book is written in an informal style to make the subject matter easily accessible. Concepts are developed with the help of motivating examples, probing questions, and many exercises. It would be suitable as a textbook for an introductory course on the topic or for selfstudy.
For this edition, more exercises and four appendices have been added.
The AMS maintains exclusive distribution rights for this edition in North America and nonexclusive distribution rights worldwide, excluding India, Pakistan, Bangladesh, Nepal, Bhutan, Sikkim, and Sri Lanka.
Graduate students and research mathematicians interested in mathematical analysis.

Chapters

Prologue. The length function

Chapter 1. Riemann integration

Chapter 2. Recipes for extending the Riemann integral

Chapter 3. General extension theory

Chapter 4. The Lebesgue measure on $\mathbb {R}$ and its properties

Chapter 5. Integration

Chapter 6. Fundamental theorem of calculus for the Lebesgue integral

Chapter 7. Measure and integration on product spaces

Chapter 8. Modes of convergence and $L_p$spaces

Chapter 9. The RadonNikodym theorem and its applications

Chapter 10. Signed measures and complex measures

Appendix A. Extended real numbers

Appendix B. Axiom of choice

Appendix C. Continuum hypotheses

Appendix D. Urysohn’s lemma

Appendix E. Singular value decomposition of a matrix

Appendix F. Functions of bounded variation

Appendix G. Differentiable transformations

From reviews for the first edition: Distinctive features include: 1) An unusually extensive treatment of the historical developments leading up to the Lebesgue integral … 2) Presentation of the standard extension of an abstract measure on an algebra to a sigma algebra prior to the final stage of development of Lebesgue measure. 3) Extensive treatment of change of variables theorems for functions of one and several variables … the conversational tone and helpful insights make this a useful introduction to the topic … The material is presented with generous details and helpful examples at a level suitable for an introductory course or for selfstudy.
Zentralblatt MATH 
A special feature [of the book] is the extensive historical and motivational discussion … At every step, whenever a new concept is introduced, the author takes pains to explain how the concept can be seen to arise naturally … The book attempts to be comprehensive and largely succeeds … The text can be used for either a onesemester or a oneyear course at M.Sc. level … The book is clearly a labor of love. The exuberance of detail, the wealth of examples and the evident delight in discussing variations and counter examples, all attest to that … All in all, the book is highly recommended to serious and demanding students.
Resonance — journal of science education