Measure Theory and Integration

Häftad, Engelska, 2006

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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.

Produktinformation

Utgivningsdatum2006-01-30
FormatHäftad
SpråkEngelska
SerieGraduate Studies in Mathematics
Antal sidor319
FörlagAmerican Mathematical Society
ISBN9781470484767

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ChaptersChapter 1. The Riemann integralChapter 2. Lebesgue measure on the lineChapter 3. Integration on measure spacesChapter 4. $L^p$ spacesChapter 5. The Caratheodory construction of measuresChapter 6. Product measuresChapter 7. Lebesgue measure on $\mathbb {R}^n$ and on manifoldsChapter 8. Signed measures and complex measuresChapter 9. $L^p$ spaces, IIChapter 10. Sobolev spacesChapter 11. Maximal functions and a.e. phenomenaChapter 12. Hausdorff's $r$-dimensional measuresChapter 13. Radon measuresChapter 14. Ergodic theoryChapter 15. Probability spaces and random variablesChapter 16. Wiener measure and Brownian motionChapter 17. Conditional expectation and martingalesAppendix A. Metric spaces, topological spaces, and compactnessAppendix B. Derivatives, diffeomorphisms, and manifoldsAppendix C. The Whitney Extension TheoremAppendix D. The Marcinkiewicz Interpolation TheoremAppendix E. Sard's TheoremAppendix F. A change of variable theorem for many-to-one mapsAppendix G. Integration of differential formsAppendix H. Change of variables revisitedAppendix 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