Hardcover ISBN:  9780821841808 
Product Code:  GSM/76 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $55.20 
Electronic ISBN:  9781470411558 
Product Code:  GSM/76.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 

Book DetailsGraduate Studies in MathematicsVolume: 76; 2006; 319 ppMSC: Primary 28;
This selfcontained treatment of measure and integration begins with a brief review of the Riemann integral and proceeds to a construction of Lebesgue measure on the real line. From there the reader is led to the general notion of measure, to the construction of the Lebesgue integral on a measure space, and to the major limit theorems, such as the Monotone and Dominated Convergence Theorems. The treatment proceeds to \(L^p\) spaces, normed linear spaces that are shown to be complete (i.e., Banach spaces) due to the limit theorems. Particular attention is paid to \(L^2\) spaces as Hilbert spaces, with a useful geometrical structure.
Having gotten quickly to the heart of the matter, the text proceeds to broaden its scope. There are further constructions of measures, including Lebesgue measure on \(n\)dimensional Euclidean space. There are also discussions of surface measure, and more generally of Riemannian manifolds and the measures they inherit, and an appendix on the integration of differential forms. Further geometric aspects are explored in a chapter on Hausdorff measure. The text also treats probabilistic concepts, in chapters on ergodic theory, probability spaces and random variables, Wiener measure and Brownian motion, and martingales.
This text will prepare graduate students for more advanced studies in functional analysis, harmonic analysis, stochastic analysis, and geometric measure theory.ReadershipGraduate students interested in analysis.

Table of Contents

Chapters

Chapter 1. The Riemann integral

Chapter 2. Lebesgue measure on the line

Chapter 3. Integration on measure spaces

Chapter 4. $L^p$ spaces

Chapter 5. The Caratheodory construction of measures

Chapter 6. Product measures

Chapter 7. Lebesgue measure on $\mathbb {R}^n$ and on manifolds

Chapter 8. Signed measures and complex measures

Chapter 9. $L^p$ spaces, II

Chapter 10. Sobolev spaces

Chapter 11. Maximal functions and a.e. phenomena

Chapter 12. Hausdorff’s $r$dimensional measures

Chapter 13. Radon measures

Chapter 14. Ergodic theory

Chapter 15. Probability spaces and random variables

Chapter 16. Wiener measure and Brownian motion

Chapter 17. Conditional expectation and martingales

Appendix A. Metric spaces, topological spaces, and compactness

Appendix B. Derivatives, diffeomorphisms, and manifolds

Appendix C. The Whitney Extension Theorem

Appendix D. The Marcinkiewicz Interpolation Theorem

Appendix E. Sard’s Theorem

Appendix F. A change of variable theorem for manytoone maps

Appendix G. Integration of differential forms

Appendix H. Change of variables revisited

Appendix I. The GaussGreen formula on Lipschitz domains


Additional Material

Reviews

Taylor's treatment throughout is elegant and very efficient … I found reading the text very enjoyable.
MAA Reviews 
The book is very understandable, requiring only a basic knowledge of analysis. It can be warmly recommended to a broad spectrum of readers, to graduate students as well as young researchers.
EMS Newsletter 
This monograph provides a quite comprehensive presentation of measure and integration theory and of some of their applications.
Mathematical Reviews


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This selfcontained treatment of measure and integration begins with a brief review of the Riemann integral and proceeds to a construction of Lebesgue measure on the real line. From there the reader is led to the general notion of measure, to the construction of the Lebesgue integral on a measure space, and to the major limit theorems, such as the Monotone and Dominated Convergence Theorems. The treatment proceeds to \(L^p\) spaces, normed linear spaces that are shown to be complete (i.e., Banach spaces) due to the limit theorems. Particular attention is paid to \(L^2\) spaces as Hilbert spaces, with a useful geometrical structure.
Having gotten quickly to the heart of the matter, the text proceeds to broaden its scope. There are further constructions of measures, including Lebesgue measure on \(n\)dimensional Euclidean space. There are also discussions of surface measure, and more generally of Riemannian manifolds and the measures they inherit, and an appendix on the integration of differential forms. Further geometric aspects are explored in a chapter on Hausdorff measure. The text also treats probabilistic concepts, in chapters on ergodic theory, probability spaces and random variables, Wiener measure and Brownian motion, and martingales.
This text will prepare graduate students for more advanced studies in functional analysis, harmonic analysis, stochastic analysis, and geometric measure theory.
Graduate students interested in analysis.

Chapters

Chapter 1. The Riemann integral

Chapter 2. Lebesgue measure on the line

Chapter 3. Integration on measure spaces

Chapter 4. $L^p$ spaces

Chapter 5. The Caratheodory construction of measures

Chapter 6. Product measures

Chapter 7. Lebesgue measure on $\mathbb {R}^n$ and on manifolds

Chapter 8. Signed measures and complex measures

Chapter 9. $L^p$ spaces, II

Chapter 10. Sobolev spaces

Chapter 11. Maximal functions and a.e. phenomena

Chapter 12. Hausdorff’s $r$dimensional measures

Chapter 13. Radon measures

Chapter 14. Ergodic theory

Chapter 15. Probability spaces and random variables

Chapter 16. Wiener measure and Brownian motion

Chapter 17. Conditional expectation and martingales

Appendix A. Metric spaces, topological spaces, and compactness

Appendix B. Derivatives, diffeomorphisms, and manifolds

Appendix C. The Whitney Extension Theorem

Appendix D. The Marcinkiewicz Interpolation Theorem

Appendix E. Sard’s Theorem

Appendix F. A change of variable theorem for manytoone maps

Appendix G. Integration of differential forms

Appendix H. Change of variables revisited

Appendix I. The GaussGreen formula on Lipschitz domains

Taylor's treatment throughout is elegant and very efficient … I found reading the text very enjoyable.
MAA Reviews 
The book is very understandable, requiring only a basic knowledge of analysis. It can be warmly recommended to a broad spectrum of readers, to graduate students as well as young researchers.
EMS Newsletter 
This monograph provides a quite comprehensive presentation of measure and integration theory and of some of their applications.
Mathematical Reviews