An introduction to mathematical analysis for economic theory and econometrics

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. To illustrate difficult concepts, the authors use simple examples drawn from economic theory and econometrics.

Accessible and rigorous, the book is self-contained, providing proofs of theorems and assuming only an undergraduate background in calculus and linear algebra.

Begins with mathematical analysis and economic examples accessible to advanced undergraduates in order to build intuition for more complex analysis used by graduate students and researchers
Takes a unified approach to understanding basic and advanced spaces of numbers through application of the Metric Completion Theorem
Focuses on examples from econometrics to explain topics in measure theory

Author Notes

Dean Corbae is the Rex A. and Dorothy B. Sebastian Centennial Professor in Business Administration at the University of Texas at Austin. Maxwell B. Stinchcombe is the E. C. McCarty Centennial Professor of Economics at the University of Texas at Austin. Juraj Zeman is researcher at the National Bank of Slovakia and lecturer in applied mathematics at Comenius University in Bratislava.

Preface	p. xi
User's Guide	p. xiii
Notation	p. xix
Chapter 1 Logic	p. 1
1.1 Statements, Sets, Subsets, and Implication	p. 1
1.2 Statements and Their Truth Values	p. 3
1.3 Proofs, a First Look	p. 6
1.4 Logical Quantifiers	p. 9
1.5 Taxonomy of Proofs	p. 11
Chapter 2 Set Theory	p. 15
2.1 Some Simple Questions	p. 16
2.2 Notation and Other Basics	p. 17
2.3 Products, Relations, Correspondences, and Functions	p. 21
2.4 Equivalence Relations	p. 26
2.5 Optimal Choice for Finite Sets	p. 28
2.6 Direct and Inverse Images, Compositions	p. 33
2.7 Weak and Partial Orders, Lattices	p. 39
2.8 Monotonic Changes in Optima: Supermodularity and Lattices	p. 42
2.9 Tarski's Lattice Fixed-Point Theorem and Stable Matchings	p. 49
2.10 Finite and Infinite Sets	p. 56
2.11 The Axiom of Choice and Some Equivalent Results	p. 62
2.12 Revealed Preference and Rationalizability	p. 64
2.13 Superstructures	p. 68
2.14 Bibliography	p. 69
2.15 End-of-Chapter Problems	p. 70
Chapter 3 The Space of Real Numbers	p. 72
3.1 Why We Want More Than the Rationals	p. 72
3.2 Basic Properties of Rationals	p. 73
3.3 Distance, Cauchy Sequences, and the Real Numbers	p. 75
3.4 The Completeness of the Real Numbers	p. 82
3.5 Examples Using Completeness	p. 87
3.6 Supremum and Infimum	p. 90
3.7 Summability	p. 92
3.8 Products of Sequences and e x	p. 99
3.9 Patience, Lim inf, and Lim sup	p. 101
3.10 Some Perspective on Completing the Rationals	p. 104
3.11 Bibliography	p. 105
Chapter 4 The Finite-Dimensional Metric Space of Real Vectors	p. 106
4.1 The Basic Definitions for Metric Spaces	p. 107
4.2 Discrete Spaces	p. 113
4.3 R l as a Normed Vector Space	p. 114
4.4 Completeness	p. 120
4.5 Closure, Convergence, and Completeness	p. 124
4.6 Separability	p. 128
4.7 Compactness in R l	p. 129
4.8 Continuous Functions on R l	p. 136
4.9 Lipschitz and Uniform Continuity	p. 143
4.10 Correspondences and the Theorem of the Maximum	p. 144
4.11 Banach's Contraction Mapping Theorem	p. 154
4.12 Connectedness	p. 167
4.13 Bibliography	p. 171
Chapter 5 Finite-Dimensional Convex Analysis	p. 172
5.1 The Basic Geometry of Convexity	p. 173
5.2 The Dual Space of R l	p. 181
5.3 The Three Degrees of Convex Separation	p. 184
5.4 Strong Separation and Neoclassical Duality	p. 186
5.5 Boundary Issues	p. 194
5.6 Concave and Convex Functions	p. 199
5.7 Separation and the Hahn-Banach Theorem	p. 209
5.8 Separation and the Kuhn-Tucker Theorem	p. 214
5.9 Interpreting Lagrange Multipliers	p. 228
5.10 Differentiability and Concavity	p. 232
5.11 Fixed-Point Theorems and General Equilibrium Theory	p. 239
5.12 Fixed-Point Theorems for Nash Equilibria and Perfect Equilibria	p. 245
5.13 Bibliography	p. 258
Chapter 6 Metric Spaces	p. 259
6.1 The Space of Compact Sets and the Theorem of the Maximum	p. 260
6.2 Spaces of Continuous Functions	p. 272
6.3 D(R), the Space of Cumulative Distribution Functions	p. 293
6.4 Approximation in C(M) when M Is Compact	p. 297
6.5 Regression Analysis as Approximation Theory	p. 304
6.6 Countable Product Spaces and Sequence Spaces	p. 311
6.7 Defining Functions Implicitly and by Extension	p. 321
6.8 The Metric Completion Theorem	p. 331
6.9 The Lebesgue Measure Space	p. 335
6.10 Bibliography	p. 343
6.11 End-of-Chapter Problems	p. 344
Chapter 7 Measure Spaces and Probability	p. 355
7.1 The Basics of Measure Theory	p. 356
7.2 Four Limit Results	p. 370
7.3 Good Sets Arguments and Measurability	p. 388
7.4 Two 0-1 Laws	p. 397
7.5 Dominated Convergence, Uniform Integrability, and Continuity of the Integral	p. 400
7.6 The Existence of Nonatomic Countably Additive Probabilities	p. 411
7.7 Transition Probabilities, Product Measures, and Fubini's Theorem	p. 423
7.8 Seriously Nonmeasurable Sets and Intergenerational Equity	p. 426
7.9 Null Sets, Completions of s-Fields, and Measurable Optima	p. 430
7.10 Convergence in Distribution and Skorohod's Theorem	p. 436
7.11 Complements and Extras	p. 440
7.12 Appendix on Lebesgue Integration	p. 448
7.13 Bibliography	p. 451
Chapter 8 The L p (ohm; F, P) and l p Spaces, p E [1, ?]	p. 452
8.1 Some Uses in Statistics and Econometrics	p. 453
8.2 Some Uses in Economic Theory	p. 456
8.3 The Basics of L p (ohm; F, P) and l p	p. 458
8.4 Regression Analysis	p. 474
8.5 Signed Measures, Vector Measures, and Densities	p. 490
8.6 Measure Space Exchange Economies	p. 498
8.7 Measure Space Games	p. 503
8.8 Dual Spaces: Representations and Separation	p. 509
8.9 Weak Convergence in L p (?, F, P), p E [1, ?)	p. 518
8.10 Optimization of Nonlinear Operators	p. 522
8.11 A Simple Case of Parametric Estimation	p. 528
8.12 Complements and Extras	p. 541
8.13 Bibliography	p. 550
Chapter 9 Probabilities on Metric Spaces	p. 551
9.1 Choice under Uncertainty	p. 551
9.2 Stochastic Processes	p. 552
9.3 The Metric Space (Delta;(M), p)	p. 553
9.4 Two Useful Implications	p. 562
9.5 Expected Utility Preferences	p. 563
9.6 The Riesz Representation Theorem for Delta;(M), M Compact	p. 567
9.7 Polish Measure Spaces and Polish Metric Spaces	p. 569
9.8 The Riesz Representation Theorem for Polish Metric Spaces	p. 571
9.9 Compactness in Delta;(M)	p. 574
9.10 An Operator Proof of the Central Limit Theorem	p. 578
9.11 Regular Conditional Probabilities	p. 583
9.12 Conditional Probabilities from Maximization	p. 589
9.13 Nonexistence of rcp's	p. 590
9.14 Bibliography	p. 594
Chapter 10 Infinite-Dimensional Convex Analysis	p. 595
10.1 Topological Spaces	p. 595
10.2 Locally Convex Topological Vector Spaces	p. 603
10.3 The Dual Space and Separation	p. 606
10.4 Filterbases, Filters, and Ultrafilters	p. 610
10.5 Bases, Subbases, Nets, and Convergence	p. 612
10.6 Compactness	p. 617
10.7 Compactness in Topological Vector Spaces	p. 621
10.8 Fixed Points	p. 624
10.9 Bibliography	p. 626
Chapter 11 Expanded Spaces	p. 627
11.1 The Basics of * R	p. 628
11.2 Superstructures, Transfer, Spillover, and Saturation	p. 632
11.3 Loeb Spaces	p. 642
11.4 Saturation, Star-Finite Maximization Models, and Compactification	p. 649
11.5 The Existence of a Purely Finitely Additive {{0, 1}}-Valued m	p. 652
11.6 Problems and Complements	p. 653
11.7 Bibliography	p. 654
Index	p. 655

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