AN INTRODUCTION TO MATHEMATICAL ANALYSIS ECONOMIC THEORY AND ECONOMETRICS

Transcription

1 AN INTRODUCTION TO MATHEMATICAL ANALYSIS FOR ECONOMIC THEORY AND ECONOMETRICS Dean Corbae Maxwell B. Stinchcombe Juraj Zeman PRINCETON UNIVERSITY PRESS Princeton and Oxford

2 Contents Preface User's Guide Notation xi xiii xix 1 Logic Statements, Sets, Subsets, and Implication Statements and Their Truth Values Proofs, a First Look Logical Quantifiers Taxonomy of Proofs 11 2 Set Theory Some Simple Questions Notation and Other Basics Products, Relations, Correspondences, and Functions Equivalence Relations Optimal Choice for Finite Sets Direct and Inverse Images, Compositions Weak and Partial Orders, Lattices Monotonic Changes in Optima: Supermodularity and Lattices Tarski's Lattice Fixed-Point Theorem and Stable Matchings Finite and Infinite Sets The Axiom of Choice and Some Equivalent Results Revealed Preference and Rationalizability Superstructures End-of- Problems 70 3 The Space of Real Numbers Why We Want More Than the Rationals Basic Properties of Rationals Distance, Cauchy Sequences, and the Real Numbers 75 vii

3 vm Contents The Completeness of the Real Numbers Examples Using Completeness Supremum and Infimum Summability Products of Sequences and e x Patience, Lim inf, and Lim sup Some Perspective on Completing the Rationals The Finite-Dimensional Metric Space of Real Vectors The Basic Definitions for Metric Spaces Discrete Spaces R^ as a Normed Vector Space Completeness Closure, Convergence, and Completeness Separability Compactness in R f Continuous Functions on M 1 Lipschitz and Uniform Continuity Correspondences and the Theorem of the Maximum Banach's Contraction Mapping Theorem Connectedness Finite-Dimensional Convex Analysis The Basic Geometry of Convexity The Dual Space of R l The Three Degrees of Convex Separation Strong Separation and Neoclassical Duality Boundary Issues Concave and Convex Functions Separation and the Hahn-Banach Theorem Separation and the Kuhn-Tucker Theorem Interpreting Lagrange Multipliers Differentiability and Concavity Fixed-Point Theorems and General Equilibrium Theory Fixed-Point Theorems for Nash Equilibria and Perfect Equilibria Metric Spaces The Space of Compact Sets and the Theorem of the Maximum Spaces of Continuous Functions D(K), the Space of Cumulative Distribution Functions Approximation in C(M) when M Is Compact Regression Analysis as Approximation Theory Countable Product Spaces and Sequence Spaces 311

4 Contents ix 6.7 Denning Functions Implicitly and by Extension The Metric Completion Theorem The Lebesgue Measure Space End-of- Problems Measure Spaces and Probability The Basics of Measure Theory Four Limit Results Good Sets Arguments and Measurability Two 0-1 Laws Dominated Convergence, Uniform Integrability, and Continuity of the Integral The Existence of Nonatomic Countably Additive Probabilities Transition Probabilities, Product Measures, and Fubini's Theorem Seriously Nonmea.surable Sets and Intergenerational Equity Null Sets, Completions of cr-fields, and Measurable Optima Convergence in Distribution and Skorohod's Theorem Complements and Extras Appendix on Lebesgue Integration The L p {Sl, J, P) and l p Spaces, p e [1, oo] Some Uses in Statistics and Econometrics Some Uses in Economic Theory The Basics of LP{Sl, "3,P) and l p Regression Analysis Signed Measures, Vector Measures, and Densities Measure Space Exchange Economies Measure Space Games Dual Spaces: Representations and Separation Weak Convergence in LP(Q, 3", P), p e [1, oo) Optimization of Nonlinear Operators A Simple Case of Parametric Estimation Complements and Extras Probabilities on Metric Spaces Choice under Uncertainty Stochastic Processes The Metric Space (A(M),p) Two Useful Implications Expected Utility Preferences The Riesz Representation Theorem for A (M), M Compact Polish Measure Spaces and Polish Metric Spaces The Riesz Representation Theorem for Polish Metric Spaces 571

5 x Contents Compactness in A(M) An Operator Proof of the Central Limit Theorem Regular Conditional Probabilities Conditional Probabilities from Maximization Nonexistence of rep's Infinite-Dimensional Convex Analysis Topological Spaces Locally Convex Topological Vector Spaces The Dual Space and Separation Filterbases, Filters, and Ultrafilters Bases, Subbases, Nets, and Convergence Compactness Compactness in Topological Vector Spaces Fixed Points 11 Expanded Spaces The Basics of *R Superstructures, Transfer, Spillover, and Saturation Loeb Spaces Saturation, Star-Finite Maximization Models, and Compactification The Existence of a Purely Finitely Additive {0, 1}-Valued \x Problems and Complements Biblioeranhv Index 655