A Course in Real Analysis

Transcription

1 A Course in Real Analysis John N. McDonald Department of Mathematics Arizona State University Neil A. Weiss Department of Mathematics Arizona State University Biographies by Carol A. Weiss New ACADEMIC PRESS San Diego London Boston York Sydney Tokyo Toronto

2 Contents Preface xiii PART ONE D Set Theory, Real Numbers, and Calculus 1 D SET THEORY Biography: Georg Cantor Basic Definitions and Properties Functions and Sets Equivalence of Sets; Countability Algebras, a-algebras, and Monotone Classes 26 2 THE REAL NUMBER SYSTEM AND CALCULUS Biography: Georg Friedrich Bernhard Riemann The Real Number System Sequences of Real Numbers 43 vii

3 viii a Contents 2.3 Open and Closed Sets Real-Valued Functions The Cantor Set and Cantor Function The Riemann Integral 81 PART TWO D Measure, Integration, and Differentiation 3 D LEBESGUE THEORY ON THE REAL LINE Biography: Emile Felix-Edouard-Justin Borel Borel Measurable Functions and Borel Sets Lebesgue Outer Measure Further Properties of Lebesgue Outer Measure Lebesgue Measure The Lebesgue Integral for Nonnegative Functions Convergence Properties of the Lebesgue Integral for Nonnegative Functions The General Lebesgue Integral Lebesgue Almost Everywhere D MEASURE THEORY Biography: Henri Leon Lebesgue Measure Spaces Measurable Functions The Abstract Lebesgue Integral for Nonnegative Functions The General Abstract Lebesgue Integral Convergence in Measure Extensions to Measures The Lebesgue-Stieltjes Integral Product Measure Spaces Iteration of Integrals in Product Measure Spaces 245

4 Contents D ix 5 a ELEMENTS OF PROBABILITY Biography: Andrei Nikolaevich Kolmogorov The Mathematical Model for Probability Random Variables Expectation of Random Variables The Law of Large Numbers a DIFFERENTIATION Biography: Johann Radon Derivatives and Dini-Derivates Functions of Bounded Variation The Indefinite Lebesgue Integral Absolutely Continuous Functions Signed Measures The Radon-Nikodym Theorem Signed and Complex Measures Decomposition of Measures Measurable Transformations and the General Change-of-Variable Formula 402 PART THREE D Topological, Metric, and Normed Spaces 7 D ELEMENTS OF TOPOLOGICAL, METRIC, AND NORMED SPACES Biography: Pavel Samuilovich Urysohn Introduction to Topological Spaces Metrics and Norms Weak Topologies Closed Sets, Convergence, and Completeness Nets and Continuity Separation Properties Connected Sets Separability, Second Countability, and Metrizability Compact Metric Spaces 464

5 x D Contents 7.10 Compact Topological Spaces Locally Compact Spaces Function Spaces D COMPLETE SPACES, COMPACT SPACES, AND APPROXIMATION Biography: Marshall Harvey Stone The Baire Category Theorem Contractions of Complete Metric Spaces Compactness in the Space C(il, Л) Compactness of Product Spaces Approximation by Functions From a Lattice Approximation by Functions From an Algebra D HILBERT SPACES AND THE CLASSICAL BANACH SPACES Biography: David Hilbert Preliminaries on Normed Spaces Hilbert Spaces Bases and Duality in Hilbert Spaces P -Spaces Nonnegative Linear Functionals on C(f2) The Dual Spaces of C(fi) and C 0 (O) D BASIC THEORY OF NORMED AND LOCALLY CONVEX SPACES Biography: Stefan Banach The Hahn-Banach Theorem Linear Operators on Banach Spaces Topological Linear Spaces Weak and Weak* Topologies Compact Convex Sets 618

6 Contents a xi PART FOUR D Harmonie Analysis and Dynamical Systems 11 D ELEMENTS OF HARMONIC ANALYSIS Biography: Ingrid Daubechies 11.1 Introduction to Fourier Series 11.2 Convergence of Fourier Series 11.3 The Fourier Transform 11.4 Fourier Transforms of Measures 11.5 С -Theory of the Fourier Transform 11.6 Introduction to Wavelets 11.7 Orthonormal Wavelet Bases; The Wavelet Transform D MEASURABLE DYNAMICAL SYSTEMS Biography: Claude Elwood Shannon Introduction and Examples Ergodic Theory Isomorphism of Measurable Dynamical Systems; Entropy The Kolmogorov-Sinai Theorem; Calculation of Entropy 723 Index 733