Book:Dean Corbae/An Introduction to Mathematical Analysis for Economic Theory and Econometrics

Subject Matter

 * Economics

Contents

 * Preface
 * User's Guide
 * Notation


 * Chapter 1: Logic
 * 1.1 Statements, Sets, Subsets and Implication
 * 1.2 Statements and Their Truth Values
 * 1.3 Proofs, a First Look
 * 1.4 Logical Quantifiers
 * 1.5 Taxonomy of Proofs


 * Chapter 2: Set Theory
 * 2.1 Some Simple Questions
 * 2.2 Notation and Other Basics
 * 2.3 Products, Relations, Correspondences, and Functions
 * 2.4 Equivalence Relations
 * 2.5 Optimal Choice for Finite Sets
 * 2.6 Direct and Inverse Images, Compositions
 * 2.7 Weak and Partial Orders, Lattices
 * 2.8 Monotonic Changes in Optima: Supermodularity and Lattices
 * 2.9 Tarski's Lattice Fixed-Point Theorem and Stable Matchings
 * 2.10 Finite and Infinite Sets
 * 2.11 The Axiom of Choice and Some Equivalent Results
 * 2.12 Revealed Preference and Rationalizability
 * 2.13 Superstructures
 * 2.14 Bibliography
 * 2.15 End-of-Chapter Problems


 * Chapter 3: The Space of Real Numbers
 * 3.1 Why We Want More Than the Rationals
 * 3.2 Basic Properties of Rationals
 * 3.3 Distance, Cauchy Sequences, and the Real Numbers
 * 3.4 The Completeness of the Real Numbers
 * 3.5 Examples Using Completeness
 * 3.6 Supremum and Infimum
 * 3.7 Summability
 * 3.8 Products of Sequences and $e^x$
 * 3.9 Patience, Lim inf, and Lim sup
 * 3.10 Some Perspective on Completing the Rationals
 * 3.11 Bibliography


 * Chapter 4: The Finite-Dimensional Metric Space of Real Vectors
 * 4.1 The Basic Definitions for Metric Spaces
 * 4.2 Discrete Spaces
 * 4.3 $\R^\ell$ as a Normed Vector Space
 * 4.4 Completeness
 * 4.5 Closure, Convergence, and Completeness
 * 4.6 Separability
 * 4.7 Compactness in $\R^\ell$
 * 4.8 Continuous Functions on $\R^\ell$
 * 4.9 Lipschitz and Uniform Continuity
 * 4.10 Correspondences and the Theorem of the Maximum
 * 4.11 Banach's Contraction Mapping Theorem
 * 4.12 Connectedness
 * 4.13 Bibliography


 * Chapter 5: Finite-Dimensional Convex Analysis
 * 5.1 The Basic Geometry of Convexity
 * 5.2 The Dual Space of $\R^\ell$
 * 5.3 The Three Degrees of Convex Separation
 * 5.4 Strong Separation and Neoclassical Duality
 * 5.5 Boundary Issues
 * 5.6 Concave and Convex Functions
 * 5.7 Separation and the Hahn-Banach Theorem
 * 5.8 Separation and the Kuhn-Tucker Theorem
 * 5.9 Interpreting Lagrange Multipliers
 * 5.10 Differentiability and Concavity
 * 5.11 Fixed-Point Theorems and General Equilibrium Theory
 * 5.12 Fixed-Point Theorems for Nash Equilibria and Perfect Equilibria
 * 5.13 Bibliography


 * Chapter 6: Metric Spaces
 * 6.1 The Space of Compact Sets and the Theorem of the Maximum
 * 6.2 Spaces of Continuous Functions
 * 6.3 $\mathcal D \left({\R}\right)$, the Space of Cumulative Distribution Functions
 * 6.4 Approximation in $C \left({M}\right)$ when $M$ Is Compact
 * 6.5 Regression Analysis as Approximation Theory
 * 6.6 Countable Product Spaces and Sequence Spaces
 * 6.7 Defining Functions Implicitly and by Extension
 * 6.8 The Metric Completion Theorem
 * 6.9 The Lebesgue Measure Space
 * 6.10 Bibliography
 * 6.11 End-of-Chapter Problems


 * Chapter 7: Measure Spaces and Probability
 * 7.1 The Basics of Measure Theory
 * 7.2 Four Limit Results
 * 7.3 Good Sets Arguments and Measurability
 * 7.4 Two $0$-$1$ Laws
 * 7.5 Dominated Convergence, Uniform Integrability, and Continuity of the Integral
 * 7.6 The Existence of Nonatomic Countably Additive Probabilities
 * 7.7 Transition Probabilities, Product Measures, and Fubini's Theorem
 * 7.8 Seriously Nonmeasurable Sets and Intergenerational Equity
 * 7.9 Null Sets, Completions of $\sigma$-Fields, and Measurable Optima
 * 7.10 Convergence in Distribution and Skorohod's Theorem
 * 7.11 Complements and Extras
 * 7.12 Appendix on Lebesgue Integration
 * 7.13 Bibliography


 * Chapter 8: The $L^p \left({\Omega, \mathcal F, P}\right)$ and $\ell^p$ spaces, $p \in \left[{1, \infty}\right]$
 * 8.1 Some Uses in Statistics and Econometrics
 * 8.2 Some Uses in Economic Theory
 * 8.3 The Basics of $L^p \left({\Omega, \mathcal F, P}\right)$ and $\ell^p$
 * 8.4 Regression Analysis
 * 8.5 Signed Measures, Vector Measures, and Densities
 * 8.6 Measure Space Exchange Economies
 * 8.7 Measure Space Games
 * 8.8 Dual Spaces: Representations and Separation
 * 8.9 Weak Convergence in $L^p \left({\Omega, \mathcal F, P}\right)$, $p \in \left[{1, \infty}\right)$
 * 8.10 Optimization of Nonlinear Operators
 * 8.11 A Simple Case of Parametric Estimation
 * 8.12 Complements and Extras
 * 8.13 Bibliography


 * Chapter 9: Probabilities on Metric Spaces
 * 9.1 Choice under Uncertainty
 * 9.2 Stochastic Processes
 * 9.3 The Metric Space $\left({\Delta \left({M}\right), \rho}\right)$
 * 9.4 Two Useful Implementations
 * 9.5 Expected Utility Preferences
 * 9.6 The Riesz Representation Theorem for $\Delta \left({M}\right)$, $M$ Compact
 * 9.7 Polish Measure Spaces and Polish Metric Spaces
 * 9.8 The Riesz Representation Theorem for Polish Metric Spaces
 * 9.9 Compactness in $\Delta \left({M}\right)$
 * 9.10 An Operator Proof of the Central Limit Theorem
 * 9.11 Regular Conditional Probabilities
 * 9.12 Conditional Probabilities from Maximization
 * 9.13 Nonexistence of rcp's
 * 9.14 Bibliography


 * Chapter 10: Infinite-Dimensional Convex Analysis
 * 10.1 Topological Spaces
 * 10.2 Locally Convex Topological Spaces
 * 10.3 The Dual Space and Separation
 * 10.4 Filterbases, Filters, and Ultrafilters
 * 10.5 Bases, Subbases, Nets, and Convergence
 * 10.6 Compactness
 * 10.7 Compactness in Topological Vector Spaces
 * 10.8 Fixed Points
 * 10.9 Bibliography


 * Chapter 11: Expanded Spaces
 * 11.1 The Basics of $* \R$
 * 11.2 Superstructures, Transfer, Spillover, and Saturation
 * 11.3 Loeb Spaces
 * 11.4 Saturation, Star-Finite Maximization Models, and Compactification
 * 11.5 The Existence of a Purely Finitely Additive $\left\{{0, 1}\right\}$-Valued $\mu$
 * 11.6 Problems and Complements
 * 11.7 Bibliography


 * Index