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 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.
* The Riemann integral * Lebesgue measure on the line * Integration on measure spaces * Lp spaces * The Caratheodory construction of measures * Product measures * Lebesgue measure on Rn and on manifolds * Signed measures and complex measures Lp spaces, II * Sobolev spaces * Maximal functions and a.e. phenomena * Hausdorff's r-dimensional measures * Radon measures * Ergodic theory * Probability spaces and random variables * Wiener measure and Brownian motion * 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 * Subject index