# 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 $2009$, **Princeton University Press**

- ISBN 978-0691118673.

### Subject Matter

### 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