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Measure Theory and Integration
 
Michael E. Taylor University of North Carolina, Chapel Hill, Chapel Hill, NC
Measure Theory and Integration
Measure Theory and Integration
Hardcover ISBN:  978-0-8218-4180-8
Product Code:  GSM/76
List Price: $99.00
MAA Member Price: $89.10
AMS Member Price: $79.20
eBook ISBN:  978-1-4704-1155-8
Product Code:  GSM/76.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-0-8218-4180-8
eBook: ISBN:  978-1-4704-1155-8
Product Code:  GSM/76.B
List Price: $184.00$141.50
MAA Member Price: $165.60$127.35
AMS Member Price: $147.20$113.20
Measure Theory and Integration
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Measure Theory and Integration
Measure Theory and Integration
Michael E. Taylor University of North Carolina, Chapel Hill, Chapel Hill, NC
Hardcover ISBN:  978-0-8218-4180-8
Product Code:  GSM/76
List Price: $99.00
MAA Member Price: $89.10
AMS Member Price: $79.20
eBook ISBN:  978-1-4704-1155-8
Product Code:  GSM/76.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN:  978-0-8218-4180-8
eBook ISBN:  978-1-4704-1155-8
Product Code:  GSM/76.B
List Price: $184.00$141.50
MAA Member Price: $165.60$127.35
AMS Member Price: $147.20$113.20
  • Book Details
     
     
    Graduate Studies in Mathematics
    Volume: 762006; 319 pp
    MSC: Primary 28;

    This self-contained 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.

    Readership

    Graduate 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 many-to-one maps
    • Appendix G. Integration of differential forms
    • Appendix H. Change of variables revisited
    • Appendix I. The Gauss-Green formula on Lipschitz domains
  • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 762006; 319 pp
MSC: Primary 28;

This self-contained 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.

Readership

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 many-to-one maps
  • Appendix G. Integration of differential forms
  • Appendix H. Change of variables revisited
  • Appendix I. The Gauss-Green 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
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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