H.S. Bear
A Primer of Lebesgue Integration, 2nd Ed.
Cover: CaseBound
Published: October 2001
ISBN: 0-12-083971-7
The Lebesgue integral is now standard for both applications and advanced mathematics. This books starts with a review of the familiar calculus integral and then constructs the Lebesgue integral from the ground up using the same ideas. A Primer of Lebesgue Integration has been used successfully both in the classroom and for individual study.
Bear presents a clear and simple introduction for those intent on further study in higher mathematics. Additionally, this book serves as a refresher providing new insight for those in the field. The author writes with an engaging, commonsense style that appeals to readers at all levels.
Preface to the 2nd Edition:
Preface The principal change from the first edition is the new one-shot definition of the Lebesgue integral. The integral is first defined for bounded functions on sets of finite measure, using upper and lower Darboux sums for finite partitions into measurable sets. This approach is designed to emphasize the similarity of the Lebesgue and Riemann integrals. By introducing countable partitions, we then extend the definition to arbitrary functions (bounded or not) and arbitrary sets (finite measure or not). This elegant touch, like many of my best ideas, was explained to me by A. M. Gleason.
Many of the errors and crudities of the first edition have been corrected, and the author is indebted to Robert Burckel, R. K. Getoor, K. P. S. Bhaskara Rao, Joel Shapiro, and Nicholas Young for pointing out assorted mistakes. In addition, several anonymous reviewers of the second edition made many helpful suggestions. I feel confident, however, that there remain enough errors to challenge and reward the conscientious reader.
Finally, the author wishes to express his gratitude to Susan Hasegawa and Pat Goldstein for their superb work with the typing and proofreading.
H.S. Bear
Contents
The Riemann's Darboux Integral
The Riemann Integral as a Limit of Sums
Lebesgue Measure on (0, 1)
Measurable Sets: The Caratheodory Characterization
The Lebesgue Integral for Bounded Functions
Properties of the Integral
The Integral of Unbounded Functions
Differentiation and Integration
Plane Measure
The Relationship between μ and
General Measures
Integration for General Measures
More Integration: The Radon?Nikodym Theorem
Product Measures
The Space L2
Inde
Bauer, Heinz
Measure and Integration Theory
Translated from the German by Robert B. Burckel
2001. 24 x 17 cm. XIIII, 236 pages. Hardcover.
ISBN 3-11-016719-0
(de Gruyter Studies in Mathematics 26)
This book gives a straightforward introduction to the field as it is nowadays required in many branches of analysis and especially in probability theory. The first three chapters (Measure Theory, Integration Theory, Product Measures) basically follow the clear and approved exposition given in the author’s earlier book on "Probability Theory and Measure Theory". Special emphasis is laid on a complete discussion of the transformation of measures and integration with respect to the product measure, convergence theorems, parameter depending integrals, as well as the Radon-Nikodym theorem.
The final chapter, essentially new and written in a clear and concise style, deals with the theory of Radon measures on Polish or locally compact spaces. With the main results being Luzin's theorem, the Riesz representation theorem, the Portmanteau theorem, and a characterization of locally compact spaces which are Polish, this chapter is a true invitation to study topological measure theory.
The text addresses graduate students, who wish to learn the fundamentals in measure and integration theory as needed in modern analysis and probability theory. It will also be an important source for anyone teaching such a course.
Contents:
Preface ・ Introduction ・ Notations ・ Measure Theory ・ Integration Theory ・ Product Measures ・ Measures on Topological Spaces ・ Bibliography ・ Index
Professor Heinz Bauer, Mathematisches Institut, Friedrich-Alexander-Universitat Erlangen-Nurnberg, Erlangen, Germany.
Anatoly N. Kochubei
Ukrainian National Academy of Sciences, Kiev, Ukraine
Pseudo-Differential Equations and Stochastics
over Non-Archimedean Fields
08/01/2001 | Hard Cover | 336 pages | Illustrated
ISBN: 0-8247-0655-2
Description
This state-of-the-art reference provides comprehensive coverage of the most recent developments in the theory of non-Archimedean pseudo-differential equations and its application to stochastics and mathematical physics?offering current methods of construction for stochastic processes on the field of p-adic numbers and related structures.
Develops a new theory for parabolic equations over non-Archimedean fields in relation to Markov processes!
>Pseudo-Differential Equations and Stochastics over Non-Archimedean Fields examines
elliptic and hyperbolic equations associated with p-adic quadratic forms
Green functions and their asymptotics
the Cauchy problem for the p-adic Schrodinger equation
spectral theory
the Fourier transform, a fractional differentiation operator, and an analog of the symmetric stable process and more!
With 1500 useful equations and over 150 contemporary references, Pseudo-Differential Equations and Stochastics over Non-Archimedean Fields is an authoritative source for pure and applied mathematicians, analysts, number theorists, mathematical physicists, and upper-level undergraduate and graduate students in these disciplines.
Table of Contents
Preliminaries
Fundamental Solutions
Spectral Theory
Parabolic Equations
Construction of Processes
Analysis over Infinite Extensions of a Local Field
Processes with Non-Archimedean Time Parameters
Bibliography
Laurent Saloff-Coste
Aspects of Sobolev-Type Inequalities
London Mathematical Society Lecture Note Series
Description
This book focuses on Poincare, Nash and other Sobolev-type inequalities and their applications to the Laplace and heat diffusion equations on Riemannian manifolds. Applications covered include the ultracontractivity of the heat diffusion semigroup, Gaussian heat kernel bounds, the Rozenblum-Lieb-Cwikel inequality and elliptic and parabolic Harnack inequalities. Emphasis is placed on the role of families of local Poincare and Sobolev inequalities. The text provides the first self contained account of the equivalence between the uniform parabolic Harnack inequality, on the one hand, and the conjunction of the doubling volume property and Poincare’s inequality on the other. It is suitable to be used as an advanced graduate textbook and will also be a useful source of information for graduate students and researchers in analysis on manifolds, geometric differential equations, Brownian motion and diffusion on manifolds, as well as other related areas.
Chapter Contents
Preface; Introduction; 1. Sobolev inequalities in Rn; 2. Moser’s elliptic Harnack Inequality; 3. Sobolev inequalities on manifolds; 4. Two applications; 5. Parabolic Harnack inequalities.
ISBN: 0-521-00607-4
Binding: Paperback
Pages: 208
available from November 2001
Ken Binmore, Joan Davies
Calculus: Concepts and Methods
Description
The pebbles used in ancient abacuses gave their name to the calculus, which today is a fundamental tool in business, economics, engineering and the sciences. This introductory book takes readers gently from single to multivariate calculus and simple differential and difference equations. Unusually the book offers a wide range of applications in business and economics, as well as more conventional scientific examples. Ideas from univariate calculus and linear algebra are covered as needed, often from a new perspective. They are reinforced in the two-dimensional case, which is studied in detail before generalisation to higher dimensions. Although there are no theorems or formal proofs, this is a serious book in which conceptual issues are explained carefully using numerous geometric devices and a wealth of worked examples, diagrams and exercises. Mathematica has been used to generate many beautiful and accurate, full-colour illustrations to help students visualise complex mathematical objects. This adds to the accessibility of the text, which will appeal to a wide audience among students of mathematics, economics and science.
Chapter Contents
Preface; Preliminaries; 1. Matrices and vectors; 2. Functions of one variable; 3. Functions of several variables; 4. Stationary points; 5. Vector-valued functions; 6. Optimization of scalar-valued functions; 7. Inverse functions; 8. Implicit functions; 9. Differentials; 10. Sums and integrals; 11. Multiple integrals; 12. Differential and difference equations of order one; 13. Complex numbers; 14. Higher order differential and difference equations.
ISBN: 0-521-77541-8
Binding: Paperback
Pages: 550
Figures: 649 line diagrams 345 exercises
available from December 2001