Real Analysis: Modern Techniques and Their Applications (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts)

An in-depth look at real analysis and its applications-now expanded and revised. This new edition of the widely used analysis book continues to cover real analysis in greater detail and at a more advanced level than most books on the subject. Encompassing several subjects that underlie much of modern analysis, the book focuses on measure and integration theory, point set topology, and the basics of functional analysis. It illustrates the use of the general theories and introduces readers to other branches of analysis such as Fourier analysis, distribution theory, and probability theory. This edition is bolstered in content as well as in scope-extending its usefulness to students outside of pure analysis as well as those interested in dynamical systems. The numerous exercises, extensive bibliography, and review chapter on sets and metric spaces make Real Analysis: Modern Techniques and Their Applications, Second Edition invaluable for students in graduate-level analysis courses. New features include: * Revised material on the n-dimensional Lebesgue integral. * An improved proof of Tychonoff's theorem. * Expanded material on Fourier analysis. * A newly written chapter devoted to distributions and differential equations. * Updated material on Hausdorff dimension and fractal dimension. Booknews Covers real analysis in greater detail and at a more advanced level than most books on the subject. Encompassing several subjects that underlie much of modern analysis, focus is on measure and integration theory, point set topology, and the basics of functional analysis. Illustrates use of the general theories and introduces readers to other branches of analysis such as Fourier analysis, distribution theory, and probability theory. This second edition offers material of interest to students outside of pure analysis as well as those interested in dynamical systems. Includes a review of sets and metric spaces, plus chapter exercises. For graduate students. Annotation c. by Book News, Inc., Portland, Or. Cover Preface Contents 0 Prologue 0.1 The Language of Set Theory 0.2 Orderings 0.3 Cardinality 0.4 More about Well Ordered Sets 0.5 The Extended Real Number System 0.6 Metric Spaces 0.7 Notes and References 1 Measures 1.1 Introduction 1.2 Sigma-algebras 1.3 Measures 1.4 Outer Measures 1.5 Borel Measures on the Real Line 1.6 Notes and References 2 Integration 2.1 Measurable Functions 2.2 Integration of Nonnegative Functions 2.3 Integration of Complex Functions 2.4 Modes of Convergence 2.5 Product Measures 2.6 The n-dimensional Lebesgue Integral 2.7 Integration in Polar Coordinates 2.8 Notes and References 3 Signed Measures and Differentiation 3.1 Signed Measures 3.2 The Lebesgue-Radon-Nikodym Theorem 3.3 Complex Measures 3.4 Differentiation on Euclidean Space 3.5 Functions of Bounded Variation 3.6 Notes and References 4 Point Set Topology 4.1 Topological Spaces 4.2 Continuous Maps 4.3 Nets 4.4 Compact Spaces 4.5 Locally Compact Hausdorff Spaces 4.6 Two Compactness Theorems 4.7 The Stone-Wierstrass Theorem 4.8 Embeddings in Cubes 4.9 Notes and References 5 Elements of Functional Analysis 5.1 Normed Vector Spaces 5.2 Linear Functionals 5.3 The Baire Category Theorem and its Consequences 5.4 Topological Vector Spaces 5.5 Hilbert Spaces 5.6 Notes and References 6 Lp Spaces 6.1 Basic Theory of Lp Spaces 6.2 The Dual of Lp 6.3 Some Useful Inequalities 6.4 Distribution Functions and Weak Lp 6.5 Interpolation of Lp Spaces 6.6 Notes and References 7 Radon Measures 7.1 Positive Linear Functionals on Cc(X) 7.2 Regularity and Approximation Theorems 7.3 The Dual of C0(X) 7.4 Products of Radon Measures 7.5 Notes and References 8 Elements of Fourier Analysis 8.1 Preliminaries 8.2 Convolutions 8.3 The Fourier Transform 8.4 Summation of Fourier Integrals and Series 8.5 Pointwise Convergence of Fourier Series 8.6 Fourier Analysis of Measures 8.7 Applications to Partial Differential Equations 8.8 Notes and References 9 Elements of Distribution Theory 9.1 Distributions 9.2 Compactly Supported, Tempered, and Periodic Distributions 9.3 Sobolev Spaces 9.4 Notes and References 10 Topics in Probability Theory 10.1 Basic Concepts 10.2 The Law of Large Numbers 10.3 The Central Limit Theorem 10.4 Construction of Sample Spaces 10.5 The Wiener Process 10.6 Notes and References 11 More Measures and Integrals 11.1 Topological Groups and Haar Measure 11.2 Hausdorff Measure 11.3 Self-similarity and Hausdorff Dimension 11.4 Integration on Manifolds 11.5 Notes and References Bibliography Index of Notation Index Many areas of mathematics utilize an imaginary variable with a real variable to form a complex number. Real analysis, as the name implies, is the section of mathematics that studies the functions of a real variable. Real variables are involved in such areas as measurements, integration, and topology. This book covers this fundamental terrain, including complete coverage of measure and integration theory, some point set topology, and rudiments of functional analysis. Real Analysis studies the functions of a real variable, including such areas as measurements and integration and topology. Over 450 exercises of varying levels are included to give readers practice in working with the ideas presented