Measure theory and integration / Michael E. Taylor.

Format:

Book

Author/Creator:

Taylor, Michael E., 1946- author.

Series:

Graduate studies in mathematics ; v. 76.

Graduate studies in mathematics, 1065-7339 ; volume 76

Language:

English

Subjects (All):

Measure theory.

Riemann integral.

Convergence.

Probabilities.

Physical Description:

1 online resource (338 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2006]

Language Note:

English

Summary:

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 $Lp$ spaces, normed linear spaces that are shown to be complete (i.e., Banach spaces) due to the limit theorems. Particular attention is paid to $L2$ 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 ofdifferential 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.

Contents:

""Cover""; ""Title""; ""Copyright""; ""Contents""; ""Introduction""; ""Chapter 1. The Riemann Integral""; ""Chapter 2. Lebesgue Measure on the Line""; ""Chapter 3. Integration on Measure Spaces""; ""Chapter 4. L[sup(p)] Spaces""; ""Chapter 5. The Caratheodory Construction of Measures""; ""Chapter 6. Product Measures""; ""Chapter 7. Lebesgue Measure on R[sup(n)] and on Manifolds""; ""Chapter 8. Signed Measures and Complex Measures""; ""Chapter 9. L[sup(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""; ""Bibliography""; ""Symbol Index""; ""A""; ""B""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""I""; ""L""; ""M""; ""S""; ""U""; ""X""; ""Subject Index""; ""A""; ""B""; ""C""; ""D""; ""E""; ""F""; ""G""; ""H""; ""I""; ""J""; ""L""; ""M""; ""N""; ""O""; ""P""; ""R""; ""S""; ""T""; ""U""; ""V""; ""W""; ""Z""; ""Back cover""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 311-313) and indexes.

Description based on print version record.

ISBN:

1-4704-1155-5