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Real Analysis Real Analysis William G. Faris May 1, 2006 ii Contents Preface xi I Sets and Functions 1 1 Logical language and mathematical proof 3 1.1 Terms, predicates and atomic formulas . 3 1.2 Formulas . 4 1.3 Restricted variables . 5 1.4 Free and bound variables . 5 1.5 Quantifier logic . 6 1.6 Natural deduction . 8 1.7 Rules for logical operations . 9 1.8 Additional rules for or and exists . 11 1.9 Strategies for natural deduction . 12 1.10 Lemmas and theorems . 14 1.11 Relaxed natural deduction . 15 1.12 Supplement: Templates . 16 1.13 Supplement: Existential hypotheses . 18 2 Sets 23 2.1 Zermelo axioms . 23 2.2 Comments on the axioms . 24 2.3 Ordered pairs and Cartesian product . 27 2.4 Relations and functions . 28 2.5 Number systems . 29 2.6 The extended real number system . 30 2.7 Supplement: Construction of number systems . 31 3 Relations, functions, dynamical Systems 35 3.1 Identity, composition, inverse, intersection . 35 3.2 Picturing relations . 36 3.3 Equivalence relations . 36 3.4 Generating relations . 36 iii iv CONTENTS 3.5 Ordered sets . 37 3.6 Functions . 37 3.7 Relations inverse to functions . 38 3.8 Dynamical systems . 38 3.9 Picturing dynamical systems . 38 3.10 Structure of dynamical systems . 39 3.11 Isomorphism of dynamical systems . 41 4 Functions, cardinal number 43 4.1 Functions . 43 4.2 Picturing functions . 44 4.3 Indexed sums and products . 44 4.4 Cartesian powers . 45 4.5 Cardinality . 45 II Order and Metric 49 5 Ordered sets and order completeness 51 5.1 Ordered sets . 51 5.2 Positivity . 52 5.3 Greatest and least; maximal and minimal . 53 5.4 Supremum and infimum; order completeness . 54 5.5 Sequences in a complete lattice . 54 5.6 Order completion . 55 5.7 The Knaster-Tarski fixed point theorem . 56 5.8 The extended real number system . 57 5.9 The Riemann integral . 57 5.10 Supplement: The Bourbaki fixed point theorem . 59 5.11 Supplement: Zorn's lemma . 61 5.12 Supplement: Ordinal numbers . 61 6 Metric spaces 65 6.1 Metric space notions . 65 6.2 Normed vector spaces . 66 6.3 Spaces of finite sequences . 66 6.4 Spaces of infinite sequences . 66 6.5 Spaces of bounded continuous functions . 68 6.6 Open and closed sets . 68 6.7 Topological spaces . 69 6.8 Continuity . 72 6.9 Uniformly equivalent metrics . 73 6.10 Sequences . 74 6.11 Supplement: Lawvere metrics and semi-continuity . 75 CONTENTS v 7 Metric spaces and metric completeness 79 7.1 Completeness . 79 7.2 Uniform equivalence of metric spaces . 81 7.3 Completion . 81 7.4 The Banach fixed point theorem . 82 7.5 Coerciveness . 83 7.6 The regulated integral . 84 8 Completely metrizable topological spaces 87 8.1 Completely metrizable spaces . 87 8.2 Closure and interior . 89 8.3 The Baire category theorem . 89 8.4 Continuous convergence . 91 8.5 Borel functions . 92 8.6 Problems . 93 9 Metric spaces and compactness 95 9.1 Total boundedness . 95 9.2 Compactness . 96 9.3 Countable product spaces . 97 9.4 The Bolzano-Weierstrass property . 97 9.5 Compactness and continuous functions . 98 9.6 The Heine-Borel property . 99 9.7 Semicontinuity . 100 9.8 Compact sets of continuous functions . 100 9.9 Embedding a Cantor space . 102 9.10 Embedding in the Hilbert cube . 102 9.11 Locally compact metrizable spaces . 103 9.12 Summary . 104 III Measure and integral 107 10 Measurable spaces 109 10.1 σ-algebras of subsets . 109 10.2 Measurable maps . 109 10.3 The Borel σ-algebra . 110 10.4 Measurable functions . 111 10.5 σ-algebras of measurable functions . 112 10.6 Borel functions . 114 10.7 Supplement: Generating sigma-algebras . 114 11 Integrals 117 11.1 Measures and integrals . 117 11.2 Image measures and image integrals . 121 vi CONTENTS 12 Elementary integrals 123 12.1 Stone vector lattices of functions . 123 12.2 Elementary integrals . 125 12.3 Dini's theorem . 126 12.4 Dini's theorem for step functions . 127 12.5 Supplement: Monotone convergence without topology . 127 13 Existence of integrals 133 13.1 The abstract Lebesgue integral: Daniell construction . 133 13.2 Stage one . 135 13.3 Stage two . 136 13.4 Extension to measurable functions . 137 13.5 Example: the Lebesgue integral . 139 13.6 Example: The expectation for coin tossing . 139 14 Uniqueness of integrals 145 14.1 The uniqueness theorem . 145 14.2 σ-finite integrals . 146 14.3 Summation . 146 14.4 Regularity . 147 14.5 Density . 148 14.6 Monotone classes . 148 14.7 Generating monotone classes . 149 14.8 Proof of the uniqueness theorem . 150 14.9 Proof of the σ-finiteness theorem . 151 14.10Supplement: Completion of an integral . 151 15 Mapping integrals 155 15.1 Comparison of integrals . 155 15.2 Probability and expectation . 155 15.3 Image integrals . 157 15.4 The Lebesgue integral . 159 15.5 The Lebesgue-Stieltjes integral . 160 15.6 The Cantor measure and the Cantor function . 162 15.7 Supplement: Direct construction of the Lebesgue-Stieltjes measure163 16 Convergence theorems 167 16.1 Convergence theorems . 167 16.2 Measure . 169 16.3 Extended real valued measurable functions . 171 16.4 Fubini's theorem for sums and integrals . 171 16.5 Fubini's theorem for sums . 172 CONTENTS vii 17 Fubini's theorem 175 17.1 Introduction . 175 17.2 Product sigma-algebras . 177 17.3 The product integral . ..
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