An Introduction to Mathenatical analysis in econonics Dean Corbae and Juraj zeman December 2002 IStill Preliminary. not to be photocopied or distributed without permission of the authors
An Introduction to Mathematical Analysis in Economics1 Dean Corbae and Juraj Zeman December 2002 1Still Preliminary. Not to be photocopied or distributed without permission of the authors
Contents 1 Introduction 1.1 Rules of logic 1.2 Taxonomy of Proofs 1.3 Bibliography for Chapter 1 2 Set Theory 2.1 Set Operations 2.1.1 Algebraic properties of set operations 2.2 Cartesian Products 2.3 Relations 2.3. 1 Equivalence relations 2.3.2 Order relations 2.4 Correspondences and Functions 2.4.1 Restrictions and extension 2.4.2 Composition of functions 2.4.3 Injections and inverses 2.4.4 Surjections and bijections 2.5 Finite and Infinite Sets 2.6 Algebras of Sets 2.7 Bibliography for Chapter 2 2.8 End of Chapter Problems 3 The Space of Real Numbers 3.1 The Field Axioms 46 3.2 The Order Axioms 3.3 The Completeness Axiom 3.4 Open and Closed Sets 3.5 Borel sets
Contents 1 Introduction 13 1.1 Rules of logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2 Taxonomy of Proofs . . . . . . . . . . . . . . . . . . . . . . . 17 1.3 Bibliography for Chapter 1 . . . . . . . . . . . . . . . . . . . . 19 2 Set Theory 21 2.1 Set Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.1 Algebraic properties of set operations . . . . . . . . . . 24 2.2 Cartesian Products . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.1 Equivalence relations . . . . . . . . . . . . . . . . . . . 25 2.3.2 Order relations . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Correspondences and Functions . . . . . . . . . . . . . . . . . 30 2.4.1 Restrictions and extensions . . . . . . . . . . . . . . . 32 2.4.2 Composition of functions . . . . . . . . . . . . . . . . . 32 2.4.3 Injections and inverses . . . . . . . . . . . . . . . . . . 33 2.4.4 Surjections and bijections . . . . . . . . . . . . . . . . 33 2.5 Finite and Infinite Sets . . . . . . . . . . . . . . . . . . . . . . 34 2.6 Algebras of Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.7 Bibliography for Chapter 2 . . . . . . . . . . . . . . . . . . . . 43 2.8 End of Chapter Problems. . . . . . . . . . . . . . . . . . . . . 44 3 The Space of Real Numbers 45 3.1 The Field Axioms . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 The Order Axioms . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 The Completeness Axiom . . . . . . . . . . . . . . . . . . . . 50 3.4 Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . . 53 3.5 Borel Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3
CONTENTS Bibilography for Chapter 3 End of Chapter Problems 4 Metric Spaces 4.1 Convergence 4.1.1Co nce of functions 4.2 Completeness 4.2.1 Completion of a metric space 4.3 Compactness 4.4 Connectedness 4.5 Normed Vector Spaces 4.5.1 Convex sets 4.5.2 A finite dimensional vector space: Rn 的70888299 4.5.3 Series 4.5.4 An infinite dimensional vector space: ep 4.6 Continuous Functions 105 4.6.1 Intermediate value theorem 108 4.6.2 Extreme value theorem 110 4.6.3 Uniform continuity 4.7 Hemicontinuous Correspondences 4.7.1 Theorem of the maximum 4.8 Fixed Points and Contraction Mappings 4.8.1 Fixed points of functions 4.8.2 Contractions 4.8.3 Fixed points of correspondences 4.9 Appendix- Proofs in Chapter 4 138 4.10 Bibilography for Chapter 4 144 4.11 End of Chapter Problems 145 5 Measure Spaces 14 5.1.1 Outer measure 5.1.2 -measurable sets 154 5. 1.3 Lebesgue meets borel 5.1.4 L-measurable mappings 159 5.2 Lebesgue Integration 170 5.2.1 Riemann integrals 170 5.2.2 Lebesgue integrals
4 CONTENTS 3.6 Bibilography for Chapter 3 . . . . . . . . . . . . . . . . . . . . 63 3.7 End of Chapter Problems. . . . . . . . . . . . . . . . . . . . . 64 4 Metric Spaces 65 4.1 Convergence ..................... . . . . . . . 68 4.1.1 Convergence of functions . . . . . . . . . . . . . . . . . 75 4.2 Completeness .................... . . . . . . . 77 4.2.1 Completion of a metric space. . . . . . . . . . . . . . . 80 4.3 Compactness .................... . . . . . . . 82 4.4 Connectedness .................... . . . . . . . 87 4.5 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 88 4.5.1 Convex sets . . . . . . . . . . . . . . . . . . . . . . . . 92 4.5.2 A finite dimensional vector space: Rn . . . . . . . . . . 93 4.5.3 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.5.4 An infinite dimensional vector space: !p . . . . . . . . . 99 4.6 Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . 105 4.6.1 Intermediate value theorem . . . . . . . . . . . . . . . 108 4.6.2 Extreme value theorem . . . . . . . . . . . . . . . . . . 110 4.6.3 Uniform continuity . . . . . . . . . . . . . . . . . . . . 111 4.7 Hemicontinuous Correspondences . . . . . . . . . . . . . . . . 113 4.7.1 Theorem of the Maximum . . . . . . . . . . . . . . . . 122 4.8 Fixed Points and Contraction Mappings . . . . . . . . . . . . 127 4.8.1 Fixed points of functions . . . . . . . . . . . . . . . . . 127 4.8.2 Contractions . . . . . . . . . . . . . . . . . . . . . . . . 130 4.8.3 Fixed points of correspondences . . . . . . . . . . . . . 132 4.9 Appendix - Proofs in Chapter 4 . . . . . . . . . . . . . . . . . 138 4.10 Bibilography for Chapter 4 . . . . . . . . . . . . . . . . . . . . 144 4.11 End of Chapter Problems . . . . . . . . . . . . . . . . . . . . 145 5 Measure Spaces 149 5.1 Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.1.1 Outer measure . . . . . . . . . . . . . . . . . . . . . . 151 5.1.2 L−measurable sets . . . . . . . . . . . . . . . . . . . . 154 5.1.3 Lebesgue meets borel . . . . . . . . . . . . . . . . . . . 158 5.1.4 L-measurable mappings . . . . . . . . . . . . . . . . . 159 5.2 Lebesgue Integration . . . . . . . . . . . . . . . . . . . . . . . 170 5.2.1 Riemann integrals . . . . . . . . . . . . . . . . . . . . . 170 5.2.2 Lebesgue integrals . . . . . . . . . . . . . . . . . . . . 172
CONTENTS 5.3 General Measure 5.3.1 Signed measures 5. 4 Examples Using Measure Theory 5.4.1 Probability Spaces 5.4.21 195 5.5 Appendix- Proofs in Chapter 5 200 5.6 Bibilography for Chapter 5 6 Function Spaces 213 6. 1 The set of bounded continuous functions 6.1.1 Completeness 216 6. 1.2 Compactness 6. 1.3 Approximation arability of C(X) 6. 1.5 Fixed point theorems 6.2 Classical Banach spaces: L 229 6.2.1 Additional Topics in Lp(X 235 6.2.2 Hilbert Spaces(L2(X)) 37 6.3 Linear operators 241 6.4 Linear functionals 6.4.1 Dual spaces 248 6.4.2 Second Dual Space 6.5 Separation Results 6.5.1 Existence of equilibrium 260 6.6 timization of Nonlinear Operators 262 6.6.1 Variational methods on infinite dimensional vector spaces 262 6.6.2 Dynamic Programming 274 6.7 Appendix- Proofs for Chapter 6 284 6.8 Bibilography for Chapter 6 297 7 Topological Spaces 7.1 Continuous Functions and Homeomorphisms 7.2 Separation Axioms 7.3 Convergence and Completeness 305
CONTENTS 5 5.3 General Measure . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.3.1 Signed Measures . . . . . . . . . . . . . . . . . . . . . 185 5.4 Examples Using Measure Theory . . . . . . . . . . . . . . . . 194 5.4.1 Probability Spaces . . . . . . . . . . . . . . . . . . . . 194 5.4.2 L1 .................... . . . . . . . . . 195 5.5 Appendix - Proofs in Chapter 5 . . . . . . . . . . . . . . . . . 200 5.6 Bibilography for Chapter 5 . . . . . . . . . . . . . . . . . . . . 211 6 Function Spaces 213 6.1 The set of bounded continuous functions . . . . . . . . . . . . 216 6.1.1 Completeness . . . . . . . . . . . . . . . . . . . . . . . 216 6.1.2 Compactness . . . . . . . . . . . . . . . . . . . . . . . 218 6.1.3 Approximation . . . . . . . . . . . . . . . . . . . . . . 221 6.1.4 Separability of C(X) . . . . . . . . . . . . . . . . . . . 227 6.1.5 Fixed point theorems . . . . . . . . . . . . . . . . . . . 227 6.2 Classical Banach spaces: Lp . . . . . . . . . . . . . . . . . . . 229 6.2.1 Additional Topics in Lp(X) . . . . . . . . . . . . . . . 235 6.2.2 Hilbert Spaces (L2(X)) . . . . . . . . . . . . . . . . . . 237 6.3 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . 241 6.4 Linear Functionals . . . . . . . . . . . . . . . . . . . . . . . . 245 6.4.1 Dual spaces . . . . . . . . . . . . . . . . . . . . . . . . 248 6.4.2 Second Dual Space . . . . . . . . . . . . . . . . . . . . 252 6.5 Separation Results . . . . . . . . . . . . . . . . . . . . . . . . 254 6.5.1 Existence of equilibrium . . . . . . . . . . . . . . . . . 260 6.6 Optimization of Nonlinear Operators . . . . . . . . . . . . . . 262 6.6.1 Variational methods on infinite dimensional vector spaces262 6.6.2 Dynamic Programming . . . . . . . . . . . . . . . . . . 274 6.7 Appendix - Proofs for Chapter 6 . . . . . . . . . . . . . . . . . 284 6.8 Bibilography for Chapter 6 . . . . . . . . . . . . . . . . . . . . 297 7 Topological Spaces 299 7.1 Continuous Functions and Homeomorphisms . . . . . . . . . . 302 7.2 Separation Axioms . . . . . . . . . . . . . . . . . . . . . . . . 303 7.3 Convergence and Completeness . . . . . . . . . . . . . . . . . 305
Acknowledgements To my family: those who put up with me in the past -Jo and Phil-and especially those who put up with me in the present- Margaret, Bethany Paul and Elena. D. c To my family JZ
Acknowledgements To my family: those who put up with me in the past - Jo and Phil - and especially those who put up with me in the present - Margaret, Bethany, Paul, and Elena. D.C. To my family. J.Z. 7