Real analysis

Measure theory, integration, weak differentiation of functions, and basic introduction to functional analysis are topics studied by virtually all graduate students in mathematics and applied sciences. Analysis: Foundations and Applications covers the core mathematical topics of the subject, and will particularly attract the reader with a more applied view. The focus of this modern text is to prepare the potential researcher to a way of thinking in applied mathematics and partial differential equations. The exposition is hands-on and accommodating to this group with little or no unnecessary abstractions This text covers the core mathematical topics of analysis and will attract the reader with a more applied view. The focus is to prepare the potential researcher to a "way of thinking" in applied mathematics and partial differential equations. The exposition is hands-on and accommodating to this group with little or no unnecessary abstractions. May be used in an introductory graduate course in analysis and measure theory, or as a preparatory text for students and researchers expecting to work in analysis, PDEs, and applied mathematics. It is a solid building block for approximation theory and probability, and it provides an excellent background for PDEs and the calculus of variations. Title ......Page 1 Contents ......Page 3 Preface ......Page 12 Acknowledgments ......Page 20 1 Countable sets ......Page 22 2 The Cantor set ......Page 23 3 Cardinality ......Page 25 3.1 Some examples ......Page 26 4 Cardinality of some infinite Cartesian products ......Page 27 5 Orderings, the maximal principle, and the axiom of choice ......Page 29 6 Well-ordering ......Page 30 Problems and Complements ......Page 32 1 Topological spaces ......Page 38 2 Urysohn's lemma ......Page 40 3 The Tietze extension theorem ......Page 42 4 Bases, axioms of countability, and product topologies ......Page 43 4.1 Product topologies ......Page 45 5 Compact topological spaces ......Page 46 5.1 Sequentially compact topological spaces ......Page 47 6 Compact subsets of RN ......Page 48 7 Continuous functions on countably compact spaces ......Page 50 8 Products of compact spaces ......Page 51 9 Vector spaces ......Page 52 9.2 Linear maps and isomorphisms ......Page 54 10 Topological vector spaces ......Page 55 10.1 Boundedness and continuity ......Page 56 12 Finite-dimensional topological vector spaces ......Page 57 12.1 Locally compact spaces ......Page 58 13 Metric spaces ......Page 59 13.1 Separation and axioms of countability ......Page 60 13.3 Pseudometrics ......Page 61 14 Metric vector spaces ......Page 62 14.1 Maps between metric spaces ......Page 63 15 Spaces of continuous functions ......Page 64 16 On the structure of a complete metric space ......Page 65 17 Compact and totally bounded metric spaces ......Page 67 17.1 Precompact subsets of X ......Page 69 Problems and Complements ......Page 70 1 Partitioning open subsets of RN ......Page 86 2 Limits of sets, characteristic functions, and si-algebras ......Page 88 3 Measures ......Page 89 3.2 Some examples ......Page 92 4 Outer measures and sequential coverings ......Page 93 4.2 The Lebesgue-Stieltjes outer measure ......Page 94 5 The Hausdorff outer measure in RN ......Page 95 6 Constructing measures from outer measures ......Page 97 7 The Lebesgue-Stieltjes measure on R ......Page 100 8 The Hausdorff measure on RN ......Page 101 9 Extending measures from semialgebras to si-algebras ......Page 103 10 Necessary and sufficient conditions for measurability ......Page 105 11 More on extensions from semialgebras to si-algebras ......Page 107 12.1 A necessary and sufficient condition of measurability ......Page 109 13 A nonmeasurable set ......Page 111 14.1 A continuous increasing function f:[0,1]->[0,1] ......Page 112 14.2 On the preimage of a measurable set ......Page 114 15 More on Borel measures ......Page 115 15.2 Regular Borel measures and Radon measures ......Page 118 16 Regular outer measures and Radon measures ......Page 119 17 Vitali coverings ......Page 120 18 The Besicovitch covering theorem ......Page 124 19 Proof of Proposition 18.2 ......Page 126 20 The Besicovitch measure-theoretical covering theorem ......Page 128 Problems and Complements ......Page 131 1 Measurable functions ......Page 144 2 The Egorov theorem ......Page 147 3 Approximating measurable functions by simple functions ......Page 149 4 Convergence in measure ......Page 151 5 Quasi-continuous functions and Lusin's theorem ......Page 154 6 Integral of simple functions ......Page 156 7 The Lebesgue integral of nonnegative functions ......Page 157 8 Fatou's lemma and the monotone convergence theorem ......Page 158 9 Basic properties of the Lebesgue integral ......Page 160 10 Convergence theorems ......Page 162 12 Product of measures ......Page 163 13 On the structure of (A*B) ......Page 165 14 The Fubini-Tonelli theorem ......Page 168 15.1 Integrals in terms of distribution functions ......Page 169 15.2 Convolution integrals ......Page 170 15.3 The Marcinkiewicz integral ......Page 171 16 Signed measures and the Hahn decomposition ......Page 172 17 The Radon-Nikodym theorem ......Page 175 18.1 The Jordan decomposition ......Page 178 18.2 The Lebesgue decomposition ......Page 180 Problems and Complements ......Page 181 1 Functions of bounded variations ......Page 192 2 Dini derivatives ......Page 194 3 Differentiating functions of bounded variation ......Page 197 4 Differentiating series of monotone functions ......Page 198 5 Absolutely continuous functions ......Page 200 6 Density of a measurable set ......Page 202 7 Derivatives of integrals ......Page 203 8 Differentiating Radon measures ......Page 205 9 Existence and measurability of D ......Page 207 9.1 Proof of Proposition 9.2 ......Page 209 10.1 Representing D for < ......Page 210 11 The Lebesgue differentiation theorem ......Page 212 11.2 Lebesgue points of an integrable function ......Page 213 12 Regular families ......Page 214 13 Convex functions ......Page 215 14 Jensen's inequality ......Page 217 15 Extending continuous functions ......Page 218 16 The Weierstrass approximation theorem ......Page 220 17 The Stone-Weierstrass theorem ......Page 221 18 Proof of the Stone-Weierstrass theorem ......Page 222 18.1 Proof of Stone's theorem ......Page 223 19 The Ascoli-Arzete theorem ......Page 224 19.1 Precompact subsets of N(E) ......Page 225 Problems and Complements ......Page 226 1 Functions in LP(E) and their norms ......Page 242 1.2 The spaces Lq for q