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

## Dean Corbae, Maxwell B. Stinchcombe and Juraj Zeman: An Introduction to Mathematical Analysis for Economic Theory and Econometrics

Published $\text {2009}$, Princeton University Press

ISBN 978-0691118673.

### 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