Introduction to Mathematical Analysis for Economic Theory and Econometrics 1st edition by Dean Corbae, Maxwell Stinchcombe, Juraj Zeman 1400833086 9781400833085

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ISBN-10 ‏ : ‎1400833086 

ISBN-13 ‏ : ‎ 9781400833085

Author: Dean Corbae, Maxwell Stinchcombe, Juraj Zeman

Providing an introduction to mathematical analysis as it applies to economic theory and econometrics, this book bridges the gap that has separated the teaching of basic mathematics for economics and the increasingly advanced mathematics demanded in economics research today. Dean Corbae, Maxwell B. Stinchcombe, and Juraj Zeman equip students with the knowledge of real and functional analysis and measure theory they need to read and do research in economic and econometric theory. Unlike other mathematics textbooks for economics, An Introduction to Mathematical Analysis for Economic Theory and Econometrics takes a unified approach to understanding basic and advanced spaces through the application of the Metric Completion Theorem. This is the concept by which, for example, the real numbers complete the rational numbers and measure spaces complete fields of measurable sets. Another of the book’s unique features is its concentration on the mathematical foundations of econometrics.

 

Introduction to Mathematical Analysis for Economic Theory and Econometrics 1st Table of contents:

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 ex

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

4.1 The Basic Definitions for Metric Spaces

4.2 Discrete Spaces

4.3 Rℓ as a Normed Vector Space

4.4 Completeness

4.5 Closure, Convergence, and Completeness

4.6 Separability

4.7 Compactness in Rℓ

4.8 Continuous Functions on Rℓ

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 Rℓ

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 D(R), the Space of Cumulative Distribution Functions

6.4 Approximation in C(M) 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 σ-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 Lp(Ω, F, P) and ℓp Spaces

8.1 Some Uses in Statistics and Econometrics

8.2 Some Uses in Economic Theory

8.3 The Basics of Lp (Ω, F, P) and ℓp

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 Lp (Ω F, P)

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 (Δ (M), ρ)

9.4 Two Useful Implications

9.5 Expected Utility Preferences

9.6 The Riesz Representation Theorem for Δ (M), 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 Δ(M)

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 Vector 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 {0, 1}-Valued μ

11.6 Problems and Complements

11.7 Bibliography

 

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