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Real Analysis

Real Analysis Author J Yeh
ISBN-10 9789814578561
Release 2014-06-11
Pages 840
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This book presents a unified treatise of the theory of measure and integration. In the setting of a general measure space, every concept is defined precisely and every theorem is presented with a clear and complete proof with all the relevant details. Counter-examples are provided to show that certain conditions in the hypothesis of a theorem cannot be simply dropped. The dependence of a theorem on earlier theorems is explicitly indicated in the proof, not only to facilitate reading but also to delineate the structure of the theory. The precision and clarity of presentation make the book an ideal textbook for a graduate course in real analysis while the wealth of topics treated also make the book a valuable reference work for mathematicians. The book is also very helpful to graduate students in statistics and electrical engineering, two disciplines that apply measure theory.

Problems and Proofs in Real Analysis

Problems and Proofs in Real Analysis Author J Yeh
ISBN-10 9789814578523
Release 2014-01-15
Pages 500
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This volume consists of the proofs of 391 problems in Real Analysis: Theory of Measure and Integration (3rd Edition). Most of the problems in Real Analysis are not mere applications of theorems proved in the book but rather extensions of the proven theorems or related theorems. Proving these problems tests the depth of understanding of the theorems in the main text. This volume will be especially helpful to those who read Real Analysis in self-study and have no easy access to an instructor or an advisor.

Measure and Integration

Measure and Integration Author Leonard F. Richardson
ISBN-10 0470501146
Release 2009-07-01
Pages 256
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A uniquely accessible book for general measure and integration, emphasizing the real line, Euclidean space, and the underlying role of translation in real analysis Measure and Integration: A Concise Introduction to Real Analysis presents the basic concepts and methods that are important for successfully reading and understanding proofs. Blending coverage of both fundamental and specialized topics, this book serves as a practical and thorough introduction to measure and integration, while also facilitating a basic understanding of real analysis. The author develops the theory of measure and integration on abstract measure spaces with an emphasis of the real line and Euclidean space. Additional topical coverage includes: Measure spaces, outer measures, and extension theorems Lebesgue measure on the line and in Euclidean space Measurable functions, Egoroff's theorem, and Lusin's theorem Convergence theorems for integrals Product measures and Fubini's theorem Differentiation theorems for functions of real variables Decomposition theorems for signed measures Absolute continuity and the Radon-Nikodym theorem Lp spaces, continuous-function spaces, and duality theorems Translation-invariant subspaces of L2 and applications The book's presentation lays the foundation for further study of functional analysis, harmonic analysis, and probability, and its treatment of real analysis highlights the fundamental role of translations. Each theorem is accompanied by opportunities to employ the concept, as numerous exercises explore applications including convolutions, Fourier transforms, and differentiation across the integral sign. Providing an efficient and readable treatment of this classical subject, Measure and Integration: A Concise Introduction to Real Analysis is a useful book for courses in real analysis at the graduate level. It is also a valuable reference for practitioners in the mathematical sciences.

Real Analysis

Real Analysis Author Elias M. Stein
ISBN-10 9781400835560
Release 2009-11-28
Pages 424
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Real Analysis is the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science. After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets. Each chapter has a series of exercises, from the relatively easy to the more complex, that are tied directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises. As with the other volumes in the series, Real Analysis is accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels. Also available, the first two volumes in the Princeton Lectures in Analysis:

Measure and Integral

Measure and Integral Author Richard L. Wheeden
ISBN-10 9781498702904
Release 2015-04-24
Pages 532
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Now considered a classic text on the topic, Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis by first developing the theory of measure and integration in the simple setting of Euclidean space, and then presenting a more general treatment based on abstract notions characterized by axioms and with less geometric content. Published nearly forty years after the first edition, this long-awaited Second Edition also: Studies the Fourier transform of functions in the spaces L1, L2, and Lp, 1 p Shows the Hilbert transform to be a bounded operator on L2, as an application of the L2 theory of the Fourier transform in the one-dimensional case Covers fractional integration and some topics related to mean oscillation properties of functions, such as the classes of Hölder continuous functions and the space of functions of bounded mean oscillation Derives a subrepresentation formula, which in higher dimensions plays a role roughly similar to the one played by the fundamental theorem of calculus in one dimension Extends the subrepresentation formula derived for smooth functions to functions with a weak gradient Applies the norm estimates derived for fractional integral operators to obtain local and global first-order Poincaré–Sobolev inequalities, including endpoint cases Proves the existence of a tangent plane to the graph of a Lipschitz function of several variables Includes many new exercises not present in the first edition This widely used and highly respected text for upper-division undergraduate and first-year graduate students of mathematics, statistics, probability, or engineering is revised for a new generation of students and instructors. The book also serves as a handy reference for professional mathematicians.

An Introduction to Measure Theory

An Introduction to Measure Theory Author Terence Tao
ISBN-10 9780821869192
Release 2011-09-14
Pages 206
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This is a graduate text introducing the fundamentals of measure theory and integration theory, which is the foundation of modern real analysis. The text focuses first on the concrete setting of Lebesgue measure and the Lebesgue integral (which in turn is motivated by the more classical concepts of Jordan measure and the Riemann integral), before moving on to abstract measure and integration theory, including the standard convergence theorems, Fubini's theorem, and the Caratheodory extension theorem. Classical differentiation theorems, such as the Lebesgue and Rademacher differentiation theorems, are also covered, as are connections with probability theory. The material is intended to cover a quarter or semester's worth of material for a first graduate course in real analysis. There is an emphasis in the text on tying together the abstract and the concrete sides of the subject, using the latter to illustrate and motivate the former. The central role of key principles (such as Littlewood's three principles) as providing guiding intuition to the subject is also emphasized. There are a large number of exercises throughout that develop key aspects of the theory, and are thus an integral component of the text. As a supplementary section, a discussion of general problem-solving strategies in analysis is also given. The last three sections discuss optional topics related to the main matter of the book.

The Theory of Measures and Integration

The Theory of Measures and Integration Author Eric M. Vestrup
ISBN-10 9780470317952
Release 2009-09-25
Pages 624
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An accessible, clearly organized survey of the basic topics of measure theory for students and researchers in mathematics, statistics, and physics In order to fully understand and appreciate advanced probability, analysis, and advanced mathematical statistics, a rudimentary knowledge of measure theory and like subjects must first be obtained. The Theory of Measures and Integration illuminates the fundamental ideas of the subject-fascinating in their own right-for both students and researchers, providing a useful theoretical background as well as a solid foundation for further inquiry. Eric Vestrup's patient and measured text presents the major results of classical measure and integration theory in a clear and rigorous fashion. Besides offering the mainstream fare, the author also offers detailed discussions of extensions, the structure of Borel and Lebesgue sets, set-theoretic considerations, the Riesz representation theorem, and the Hardy-Littlewood theorem, among other topics, employing a clear presentation style that is both evenly paced and user-friendly. Chapters include: * Measurable Functions * The Lp Spaces * The Radon-Nikodym Theorem * Products of Two Measure Spaces * Arbitrary Products of Measure Spaces Sections conclude with exercises that range in difficulty between easy "finger exercises"and substantial and independent points of interest. These more difficult exercises are accompanied by detailed hints and outlines. They demonstrate optional side paths in the subject as well as alternative ways of presenting the mainstream topics. In writing his proofs and notation, Vestrup targets the person who wants all of the details shown up front. Ideal for graduate students in mathematics, statistics, and physics, as well as strong undergraduates in these disciplines and practicing researchers, The Theory of Measures and Integration proves both an able primary text for a real analysis sequence with a focus on measure theory and a helpful background text for advanced courses in probability and statistics.

Lectures on Real Analysis

Lectures on Real Analysis Author J. Yeh
ISBN-10 981023936X
Release 2000
Pages 548
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The theory of the Lebesgue integral is a main pillar in the foundation of modern analysis and its applications, including probability theory. This volume shows how and why the Lebesgue integral is such a universal and powerful concept. The lines of development of the theory are made clear by the order in which the main theorems are presented. Frequent references to earlier theorems made in the proofs emphasize the interdependence of the theorems and help to show how the various definitions and theorems fit together. Counter-examples are included to show why a hypothesis in a theorem cannot be dropped. The book is based upon a course on real analysis which the author has taught. It is particularly suitable for a one-year course at the graduate level. Precise statements and complete proofs are given for every theorem, with no obscurity left. For this reason the book is also suitable for self-study.

A User Friendly Introduction to Lebesgue Measure and Integration

A User Friendly Introduction to Lebesgue Measure and Integration Author Gail S. Nelson
ISBN-10 9781470421991
Release 2015-11-09
Pages 221
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A User-Friendly Introduction to Lebesgue Measure and Integration provides a bridge between an undergraduate course in Real Analysis and a first graduate-level course in Measure Theory and Integration. The main goal of this book is to prepare students for what they may encounter in graduate school, but will be useful for many beginning graduate students as well. The book starts with the fundamentals of measure theory that are gently approached through the very concrete example of Lebesgue measure. With this approach, Lebesgue integration becomes a natural extension of Riemann integration. Next, -spaces are defined. Then the book turns to a discussion of limits, the basic idea covered in a first analysis course. The book also discusses in detail such questions as: When does a sequence of Lebesgue integrable functions converge to a Lebesgue integrable function? What does that say about the sequence of integrals? Another core idea from a first analysis course is completeness. Are these -spaces complete? What exactly does that mean in this setting? This book concludes with a brief overview of General Measures. An appendix contains suggested projects suitable for end-of-course papers or presentations. The book is written in a very reader-friendly manner, which makes it appropriate for students of varying degrees of preparation, and the only prerequisite is an undergraduate course in Real Analysis.

Lebesgue Integration on Euclidean Space

Lebesgue Integration on Euclidean Space Author Frank Jones
ISBN-10 0763717088
Release 2001
Pages 588
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Lebesgue Integration on Euclidean Space contains a concrete, intuitive, and patient derivation of Lebesgue measure and integration on Rn. Throughout the text, many exercises are incorporated, enabling students to apply new ideas immediately. Jones strives to present a slow introduction to Lebesgue integration by dealing with n-dimensional spaces from the outset. In addition, the text provides students a thorough treatment of Fourier analysis, while holistically preparing students to become workers in real analysis.

Real Analysis

Real Analysis Author Jewgeni H. Dshalalow
ISBN-10 9781420036893
Release 2000-09-28
Pages 584
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Designed for use in a two-semester course on abstract analysis, REAL ANALYSIS: An Introduction to the Theory of Real Functions and Integration illuminates the principle topics that constitute real analysis. Self-contained, with coverage of topology, measure theory, and integration, it offers a thorough elaboration of major theorems, notions, and constructions needed not only by mathematics students but also by students of statistics and probability, operations research, physics, and engineering. Structured logically and flexibly through the author's many years of teaching experience, the material is presented in three main sections: Part 1, chapters 1through 3, covers the preliminaries of set theory and the fundamentals of metric spaces and topology. This section can also serves as a text for first courses in topology. Part II, chapter 4 through 7, details the basics of measure and integration and stands independently for use in a separate measure theory course. Part III addresses more advanced topics, including elaborated and abstract versions of measure and integration along with their applications to functional analysis, probability theory, and conventional analysis on the real line. Analysis lies at the core of all mathematical disciplines, and as such, students need and deserve a careful, rigorous presentation of the material. REAL ANALYSIS: An Introduction to the Theory of Real Functions and Integration offers the perfect vehicle for building the foundation students need for more advanced studies.

A Guide to Advanced Real Analysis

A Guide to Advanced Real Analysis Author G. B. Folland
ISBN-10 0883853434
Release 2009-11-30
Pages 107
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A concise guide to the core material in a graduate level real analysis course.

Measure Integration and Functional Analysis

Measure  Integration  and Functional Analysis Author Robert B. Ash
ISBN-10 9781483265100
Release 2014-05-10
Pages 300
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Measure, Integration, and Functional Analysis deals with the mathematical concepts of measure, integration, and functional analysis. The fundamentals of measure and integration theory are discussed, along with the interplay between measure theory and topology. Comprised of four chapters, this book begins with an overview of the basic concepts of the theory of measure and integration as a prelude to the study of probability, harmonic analysis, linear space theory, and other areas of mathematics. The reader is then introduced to a variety of applications of the basic integration theory developed in the previous chapter, with particular reference to the Radon-Nikodym theorem. The third chapter is devoted to functional analysis, with emphasis on various structures that can be defined on vector spaces. The final chapter considers the connection between measure theory and topology and looks at a result that is a companion to the monotone class theorem, together with the Daniell integral and measures on topological spaces. The book concludes with an assessment of measures on uncountably infinite product spaces and the weak convergence of measures. This book is intended for mathematics majors, most likely seniors or beginning graduate students, and students of engineering and physics who use measure theory or functional analysis in their work.

Measure Theory and Integration Second Edition

Measure Theory and Integration  Second Edition Author M.M. Rao
ISBN-10 0824754018
Release 2004-01-30
Pages 792
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Significantly revised and expanded, this authoritative reference/text comprehensively describes concepts in measure theory, classical integration, and generalized Riemann integration of both scalar and vector types-providing a complete and detailed review of every aspect of measure and integration theory using valuable examples, exercises, and applications. With more than 170 references for further investigation of the subject, this Second Edition provides more than 60 pages of new information, as well as a new chapter on nonabsolute integrals contains extended discussions on the four basic results of Banach spaces presents an in-depth analysis of the classical integrations with many applications, including integration of nonmeasurable functions, Lebesgue spaces, and their properties details the basic properties and extensions of the Lebesgue-Carathéodory measure theory, as well as the structure and convergence of real measurable functions covers the Stone isomorphism theorem, the lifting theorem, the Daniell method of integration, and capacity theory Measure Theory and Integration, Second Edition is a valuable reference for all pure and applied mathematicians, statisticians, and mathematical analysts, and an outstanding text for all graduate students in these disciplines.

Measure and Integration Theory

Measure and Integration Theory Author Heinz Bauer
ISBN-10 3110167190
Release 2001-01
Pages 230
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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.

Measure theory and Integration

Measure theory and Integration Author G De Barra
ISBN-10 9780857099525
Release 2003-07-01
Pages 240
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This text approaches integration via measure theory as opposed to measure theory via integration, an approach which makes it easier to grasp the subject. Apart from its central importance to pure mathematics, the material is also relevant to applied mathematics and probability, with proof of the mathematics set out clearly and in considerable detail. Numerous worked examples necessary for teaching and learning at undergraduate level constitute a strong feature of the book, and after studying statements of results of the theorems, students should be able to attempt the 300 problem exercises which test comprehension and for which detailed solutions are provided. Approaches integration via measure theory, as opposed to measure theory via integration, making it easier to understand the subject Includes numerous worked examples necessary for teaching and learning at undergraduate level Detailed solutions are provided for the 300 problem exercises which test comprehension of the theorems provided

Real Analysis and Probability

Real Analysis and Probability Author Robert P. Ash
ISBN-10 9781483216188
Release 2014-05-10
Pages 44
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Real Analysis and Probability: Solutions to Problems presents solutions to problems in real analysis and probability. Topics covered range from measure and integration theory to functional analysis and basic concepts of probability; the interplay between measure theory and topology; conditional probability and expectation; the central limit theorem; and strong laws of large numbers in terms of martingale theory. Comprised of eight chapters, this volume begins with problems and solutions for the theory of measure and integration, followed by various applications of the basic integration theory. Subsequent chapters deal with functional analysis, paying particular attention to structures that can be defined on vector spaces; the connection between measure theory and topology; basic concepts of probability; and conditional probability and expectation. Strong laws of large numbers are also taken into account, first from the classical viewpoint, and then via martingale theory. The final chapter is devoted to the one-dimensional central limit problem, with emphasis on the fundamental role of Prokhorov's weak compactness theorem. This book is intended primarily for students taking a graduate course in probability.