Real Analysis: Part I William G. Faris February 2, 2004 ii Contents 1 Mathematical proof 1 1.1 Logical language . 1 1.2 Free and bound variables . 3 1.3 Proofs from analysis . 4 1.4 Natural deduction . 6 1.5 Natural deduction strategies . 13 1.6 Equality . 16 1.7 Lemmas and theorems . 18 1.8 More proofs from analysis . 19 2 Sets 21 2.1 Zermelo axioms . 21 2.2 Comments on the axioms . 22 2.3 Ordered pairs and Cartesian product . 25 2.4 Relations and functions . 26 2.5 Number systems . 28 3 Relations, Functions, Dynamical Systems 31 3.1 Identity, composition, inverse, intersection . 31 3.2 Picturing relations . 32 3.3 Equivalence relations . 32 3.4 Generating relations . 32 3.5 Ordered sets . 33 3.6 Functions . 33 3.7 Relations inverse to functions . 34 3.8 Dynamical systems . 34 3.9 Picturing dynamical systems . 35 3.10 Structure of dynamical systems . 35 4 Functions, Cardinal Number 39 4.1 Functions . 39 4.2 Picturing functions . 40 4.3 Indexed sums and products . 40 4.4 Cartesian powers . 41 iii iv CONTENTS 4.5 Cardinality . 41 5 Ordered sets and completeness 45 5.1 Ordered sets . 45 5.2 Order completeness . 46 5.3 Sequences in a complete lattice . 47 5.4 Order completion . 48 5.5 The Knaster-Tarski fixed point theorem . 49 5.6 The extended real number system . 49 6 Metric spaces 51 6.1 Metric space notions . 51 6.2 Normed vector spaces . 51 6.3 Spaces of finite sequences . 52 6.4 Spaces of infinite sequences . 52 6.5 Spaces of bounded continuous functions . 54 6.6 Open and closed sets . 54 6.7 Continuity . 55 6.8 Uniformly equivalent metrics . 57 6.9 Sequences . 58 7 Metric spaces and completeness 61 7.1 Completeness . 61 7.2 Uniform equivalence of metric spaces . 63 7.3 Completion . 63 7.4 The Banach fixed point theorem . 64 7.5 Coerciveness . 65 8 Metric spaces and compactness 67 8.1 Total boundedness . 67 8.2 Compactness . 68 8.3 Countable product spaces . 69 8.4 Compactness and continuous functions . 70 8.5 Semicontinuity . 71 8.6 Compact sets of continuous functions . 71 8.7 Curves of minimum length . 73 9 Vector lattices 75 9.1 Positivity . 75 9.2 Integration of regulated functions . 76 9.3 The Riemann integral . 76 9.4 Step functions . 77 9.5 Coin tossing . 79 9.6 Vector lattices . 81 9.7 Elementary integrals . 82 9.8 Integration on a product of finite spaces . 82 CONTENTS v 10 The integral 85 10.1 The Daniell construction . 85 10.2 Stage one . 87 10.3 Stage two . 88 10.4 Example: Coin tossing . 88 10.5 Example: Lebesgue measure . 91 11 Measurable functions 95 11.1 Monotone classes . 95 11.2 Generating monotone classes . 96 11.3 Sigma-algebras of functions . 97 11.4 Generating sigma-algebras . 99 11.5 Sigma-rings of functions . 100 11.6 Rings and algebras of sets . 102 12 The integral on measurable functions 105 12.1 Integration . 105 12.2 Uniqueness of integrals . 107 12.3 Existence of integrals . 108 12.4 Probability and expectation . 108 12.5 Image integrals . 109 12.6 The Lebesgue integral . 111 12.7 The Lebesgue-Stieltjes integral . 112 12.8 Integrals on a σ-ring . 116 13 Integrals and measures 117 13.1 Terminology . 117 13.2 Convergence theorems . 118 13.3 Measure . 120 13.4 Extended real valued measurable functions . 122 13.5 Fubini's theorem for sums and integrals . 122 13.6 Fubini's theorem for sums . 123 14 Fubini's theorem 127 14.1 Introduction . 127 14.2 Sigma-finite integrals . 129 14.3 Summation . 130 14.4 Product sigma-algebras . 131 14.5 The product integral . 132 14.6 Tonelli's theorem . 133 14.7 Fubini's theorem . 136 14.8 Semirings and rings of sets . 137 vi CONTENTS 15 Probability 139 15.1 Coin-tossing . 139 15.2 Weak law of large numbers . 140 15.3 Strong law of large numbers . 141 15.4 Random walk . 142 Chapter 1 Mathematical proof 1.1 Logical language There are many useful ways to present mathematics; sometimes a picture or a physical analogy produces more understanding than a complicated equation. However, the language of mathematical logic has a unique advantage: it gives a standard form for presenting mathematical truth. If there is doubt about whether a mathematical formulation is clear or precise, this doubt can be re- solved by converting to this format. The value of a mathematical discovery is considerably enhanced if it is presented in a way that makes it clear that the result and its proof could be stated in such a rigorous framework. Here is a somewhat simplified model of the language of mathematical logic. There may be function symbols. These may be 0-place function symbols, or constants. These stand for objects in some set. Example: 8. Or they may be 1-place functions symbols. These express functions from some set to itself, that is, with one input and one output. Example: square. Or they may be 2-place function symbols. These express functions with two inputs and one output. Example: +. Once the function symbols have been specified, then one can form terms. The language also has a collection of variables x; y; z; x0; y0; z0;:::. Each variable is a term. Each constant c is a term. If t is a term, and f is a 1-place function symbol, then f(t) is a term. If s and t are terms, and g is a 2-place function symbol, then g(s; t) or (sgt) is a term. Example: In an language with constant terms 1; 2; 3 and 2-place function symbol + the expression (x + 2) is a term, and the expression (3+(x+2)) is a term. Note: Sometimes it is a convenient abbreviation to omit outer parentheses. Thus 3 + (x + 2) would be an abbreviation for (3 + (x + 2)). The second ingredient is predicate symbols. These may be 0-place predicate symbols, or propositional symbols. They may stand for complete sentences. One useful symbol of this nature is ?, which is interpreted as always false. Or they may be 1-place predicate symbols. These express properties. Example: even. 1 2 CHAPTER 1. MATHEMATICAL PROOF Or they may be 2-place predicate symbols. These express relations. Example: <. Once the terms have been specified, then the atomic formulas are specified. A propositional symbol is an atomic formula. If p is a property symbol, and t is a term, then tp is an atomic formula. If s and t are terms, and r is a relation symbol, then srt is an atomic formula. Thus (x + 2) < 3 is an atomic formula. Note: This could be abbreviated x + 2 < 3. Finally there are logical symbols ^, _, ), 8, 9, and parentheses. Once the atomic formulas are specified, then the other formulas are obtained by logical operations. If A and B are formulas, then so are (A ^ B), (A _ B), and (A ) B). If x is a variable and A(x) is a formula, then so are 8x A(x) and 9x A(x). Thus 9x x + 2 < 3 is a formula. We shall often abbreviate (A )?) by :A. Thus facts about negation will be special cases of facts about implication. In writing a formula, we often omit the outermost parentheses. However this is just an abbreviation. Another useful abbreviation is (A , B) for ((A ) B) ^ (B ) A)). Some of the logical operations deserve special comment. The implication A ) B is also written if A, then B A only if B B if A. The equivalence A , B is.