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Preface vii
Prologue 1 (18)
The Language of Set Theory 1 (3)
Orderings 4 (2)
Cardinality 6 (3)
More about Well Ordered Sets 9 (1)
The Extended Real Number System 10 (3)
Metric Spaces 13 (3)
Notes and References 16 (3)
Measures 19 (24)
Introduction 19 (2)
σ-algebras 21 (3)
Measures 24 (4)
Outer Measures 28 (5)
Borel Measures on the Real Line 33 (7)
Notes and References 40 (3)
Integration 43 (42)
Measurable Functions 43 (6)
Integration of Nonnegative Functions 49 (3)
Integration of Complex Functions 52 (8)
Modes of Convergence 60 (4)
Product Measures 64 (6)
The n-dimensional Lebesgue Integral 70 (7)
Integration in Polar Coordinates 77 (4)
Notes and References 81 (4)
Signed Measures and Differentiation 85 (28)
Signed Measures 85 (3)
The Lebesgue-Radon-Nikodym Theorem 88 (5)
Complex Measures 93 (2)
Differentiation on Euclidean Space 95 (5)
Functions of Bounded Variation 100(9)
Notes and References 109(4)
Point Set Topology 113(38)
Topological Spaces 113(6)
Continuous Maps 119(6)
Nets 125(3)
Compact Spaces 128(3)
Locally Compact Hausdorff Spaces 131(5)
Two Compactness Theorems 136(2)
The Stone-Weierstrass Theorem 138(5)
Embeddings in Cubes 143(3)
Notes and References 146(5)
Elements of Functional Analysis 151(30)
Normed Vector Spaces 151(6)
Linear Functionals 157(4)
The Baire Category Theorem and its 161(4)
Consequences
Topological Vector Spaces 165(6)
Hilbert Spaces 171(8)
Notes and References 179(2)
Lp Spaces 181(30)
Basic Theory of Lp Spaces 181(7)
The Dual of Lp 188(5)
Some Useful Inequalities 193(4)
Distribution Functions and Weak Lp 197(3)
Interpolation of Lp Spaces 200(8)
Notes and References 208(3)
Radon Measures 211(24)
Positive Linear Functionals on Cc (X) 211(5)
Regularity and Approximation Theorems 216(5)
The Dual of C0(X) 221(5)
Products of Radon Measures 226(5)
Notes and References 231(4)
Elements of Fourier Analysis 235(46)
Preliminaries 235(4)
Convolutions 239(8)
The Fourier Transform 247(10)
Summation of Fourier Integrals and Series 257(6)
Pointwise Convergence of Fourier Series 263(7)
Fourier Analysis of Measures 270(3)
Applications to Partial Differential 273(5)
Equations
Notes and References 278(3)
Elements of Distribution Theory 281(32)
Distributions 281(10)
Compactly Supported, Tempered, and Periodic 291(10)
Distributions
Sobolev Spaces 301(9)
Notes and References 310(3)
Topics in Probability Theory 313(26)
Basic Concepts 313(7)
The Law of Large Numbers 320(5)
The Central Limit Theorem 325(3)
Construction of Sample Spaces 328(2)
The Wiener Process 330(6)
Notes and References 336(3)
More Measures and Integrals 339(26)
Topological Groups and Haar Measure 339(9)
Hausdorff Measure 348(7)
Self-similarity and Hausdorff Dimension 355(6)
Integration on Manifolds 361(2)
Notes and References 363(2)
Bibliography 365(12)
Index of Notation 377(2)
Index 379

* Updated material on Hausdorff dimension and fractal dimension.