Measure Theory and Integration (Graduate Studies in Mathematics)

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. This Book Is A Survey Of Asymptotic Methods Set In The Current Applied Research Context Of Wave Propagation. It Stresses Rigorous Analysis In Addition To Formal Manipulations. Asymptotic Expansions Developed In The Text Are Justified Rigorously, And Students Are Shown How To Obtain Solid Error Estimates For Asymptotic Formulae. The Book Relates Examples And Exercises To Subjects Of Current Research Interest, Such As The Problem Of Locating The Zeros Of Taylor Polynomials Of Entire Nonvanishing Functions And The Problem Of Counting Integer Lattice Points In Subsets Of The Plane With Various Geometrical Properties Of The Boundary. The Book Is Intended For A Beginning Graduate Course On Asymptotic Analysis In Applied Mathematics And Is Aimed At Students Of Pure And Applied Mathematics As Well As Science And Engineering. The Basic Prerequisite Is A Background In Differential Equations, Linear Algebra, Advanced Calculus, And Complex Variables At The Level Of Introductory Undergraduate Courses On These Subjects.--book Jacket. Chapter 0. Themes Of Asymptotic Analysis Chapter 1. The Nature Of Asymptotic Approximations Chapter 2. Fundamental Techniques For Integrals Chapter 3. Laplace's Method For Asymptotic Expansions Of Integrals Chapter 4. The Method Of Steepest Descents For Asymptotic Expansions Of Integrals Chapter 5. The Method Of Stationary Phase For Asymptotic Analysis Of Oscillatory Integrals Chapter 6. Asymptotic Behavior Of Solutions Of Linear Second-order Differential Equations In The Complex Plane Chapter 7. Introduction To Asymptotics Of Solutions Of Ordinary Differential Equations With Respect To Parameters Chapter 8. Asymptotics Of Linear Boundary-value Problems Chapter 9. Asymptotics Of Oscillatory Phenomena Chapter 10. Weakly Nonlinear Waves Appendix: Fundamental Inequalities Peter D. Miller. Includes Bibliographical References (p. 453-454) And Indexes. This book is a survey of asymptotic methods set in the current applied research context of wave propagation. It stresses rigorous analysis in addition to formal manipulations. Asymptotic expansions developed in the text are justified rigorously, and students are shown how to obtain solid error estimates for asymptotic formulae. The book relates examples and exercises to subjects of current research interest, such as the problem of locating the zeros of Taylor polynomials of entire nonvanishing functions and the problem of counting integer lattice points in subsets of the plane with various geometrical properties of the boundary. The book is intended for a beginning graduate course on asymptotic analysis in applied mathematics and is aimed at students of pure and applied mathematics as well as science and engineering. The basic prerequisite is a background in differential equations, linear algebra, advanced calculus, and complex variables at the level of introductory undergraduate courses on these subjects. The book is ideally suited to the needs of a graduate student who, on the one hand, wants to learn basic applied mathematics, and on the other, wants to understand what is needed to make the various arguments rigorous. Down here in the Village, this is known as the Courant point of view!! —Percy Deift, Courant Institute, New York Peter D. Miller is an associate professor of mathematics at the University of Michigan at Ann Arbor. He earned a Ph.D. in Applied Mathematics from the University of Arizona and has held positions at the Australian National University (Canberra) and Monash University (Melbourne). His current research interests lie in singular limits for integrable systems. Ch. 1. Riemann Integral -- Ch. 2. Lebesgue Measure On The Line -- Ch. 3. Integration On Measure Spaces -- Ch. 4. L[superscript P] Spaces -- Ch. 5. Caratheodory Construction Of Measures -- Ch. 6. Product Measures -- Ch. 7. Lebesgue Measure On R[superscript N] And On Manifolds -- Ch. 8. Signed Measures And Complex Measures -- Ch. 9. L[superscript P] Spaces, Ii -- Ch. 10. Sobolev Spaces -- Ch. 11. Maximal Functions And A.e. Phenomena -- Ch. 12. Hansdorff's R-dimensional Measures -- Ch. 13. Radon Measures -- Ch. 14. Ergodic Theory -- Ch. 15. Probability Spaces And Random Variables -- Ch. 16. Wiener Measure And Brownian Motion -- Ch. 17. Conditional Expectation And Martingales -- App. A. Metric Spaces, Topological Spaces, And Compactness -- App. B. Derivatives, Diffeomorphisms, And Manifolds -- App. C. Whitney Extension Theorem -- App. D. Marcinkiewicz Interpolation Theorem -- App. E. Sard's Theorem -- App. F. Change Of Variable Theorem For Many-to-one Maps -- App. G. Integration Of Differential Forms -- App. H. Change Of Variables Revisited -- App. I. Gauss-green Formula On Lipschitz Domains. Michael E. Taylor. Includes Bibliographical References (p. 311-313) And Indexes. Offers a treatment of measure and integration that begins with a brief review of the Riemann integral and proceeds to a construction of Lebesgue measure on the real line. This text also treats probabilistic concepts, in chapters on ergodic theory, probability spaces and random variables, Wiener measure and Brownian motion, and martingales