**Graduate Studies in Mathematics**

Volume: 76;
2006;
319 pp;
Hardcover

MSC: Primary 28;

Print ISBN: 978-0-8218-4180-8

Product Code: GSM/76

List Price: $65.00

Individual Member Price: $52.00

**Electronic ISBN: 978-1-4704-1155-8
Product Code: GSM/76.E**

List Price: $65.00

Individual Member Price: $52.00

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#### Supplemental Materials

# Measure Theory and Integration

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*Michael E. Taylor*

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.

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## Measure Theory and Integration

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Introduction vii8 free
- Chapter 1. The Riemann Integral 116 free
- Chapter 2. Lebesgue Measure on the Line 1328
- Chapter 3. Integration on Measure Spaces 2540
- Chapter 4. L[sup(p)] Spaces 4156
- Chapter 5. The Caratheodory Construction of Measures 5772
- Chapter 6. Product Measures 7186
- Chapter 7. Lebesgue Measure on R[sup(n)] and on Manifolds 8398
- Chapter 8. Signed Measures and Complex Measures 107122
- Chapter 9. L[sup(p)] Spaces, II 113128
- Chapter 10. Sobolev Spaces 129144
- Chapter 11. Maximal Functions and A.E. Phenomena 139154
- Chapter 12. Hausdorff's r-Dimensional Measures 157172
- Chapter 13. Radon Measures 179194
- Chapter 14. Ergodic Theory 193208
- Chapter 15. Probability Spaces and Random Variables 207222
- Chapter 16. Wiener Measure and Brownian Motion 221236
- Chapter 17. Conditional Expectation and Martingales 233248
- Appendix A. Metric Spaces, Topological Spaces, and Compactness 251266
- Appendix B. Derivatives, Diffeomorphisms, and Manifolds 267282
- Appendix C. The Whitney Extension Theorem 277292
- Appendix D. The Marcinkiewicz Interpolation Theorem 283298
- Appendix E. Sard's Theorem 287302
- Appendix F. A Change of Variable Theorem for Many-to-one Maps 289304
- Appendix G. Integration of Differential Forms 293308
- Appendix H. Change of Variables Revisited 303318
- Appendix I. The Gauss-Green Formula on Lipschitz Domains 309324
- Bibliography 311326
- Symbol Index 315330
- Subject Index 317332
- Back cover Back Cover1338