Books
Real Analysis with Economic Applications
Efe A. Ok
Book Description and Endorsements
Chapter A: Preliminaries of Real Analysis
Addenda
Corrections: Typos
Chapter B: Countability
Addenda: Section B.4.3 (Rewritten)
Corrections: Typos
Chapter C: Metric Spaces
Addenda
Corrections: Typos
Chapter D: Continuity I
Addenda: The Ekeland Variational Principle / Proof of Brouwer's Fixed Point Theorem / Motzkin's Characterization of Convex Sets
Corrections: The Thoughtful Correction of Footnote 47 by Douglas Bridges / Typos
Chapter E: Continuity II
Addenda
Corrections
Chapter F: Linear Spaces
Addenda
Corrections: Typos
Chapter G: Convexity
Addenda
Corrections: Typos
Chapter H: Economic Applications
Addenda
Corrections
Chapter I: Metric Linear Spaces
Addenda
Corrections: Typos
Chapter J: Normed Linear Spaces
Addenda
Corrections
Chapter K: Differential Calculus
Addenda: On the Existence of Approximate Stationary Points
Corrections: Typos
Hints to Selected Exercises
Addenda
Corrections: Correction to Exercise F.3
Probability Theory and Measure with Economic Applications
Efe A. Ok
Preface (TBW)
Elements of Set Theory / The Real Number System / Countability / The Cantor Set / The Vitali Paradox
Chapter B: Probability via Measure Theory
Measurable Spaces / (Borel) Probability Spaces / Constructions of Probability Spaces (Coin Toss Space, Markov Chains, etc.) / The Lebesgue Measure / The Sierpinski Class Lemma
Measurable Functions / Transformations and Approximations of Random Variables / The Doob-Dynkin Lemma / Distribution of a Random Variable
Chapter D: Expectation via the Lebesgue Integral
Expectation Functional / The Lebesgue Integral / Absolute Continuity / Uniform Integrability / Expectation of Banach Space-Valued Random Variables / Application: Stochastic Dominance / Elementary Inequalities / Lp Spaces / Lp-Approximation by Lipschitz Maps / The Riesz-Markov-Kakutani Representation Theorem / Choquet's Theorem
Chapter E: Expectation via the Stieltjes Integral
The Stieltjes Integral / The Riemann Integral / The Lebesgue Criterion / Expectation as a Stieltjes Integral / Integration by Parts / Application: More on Stochastic Dominance / Application: Portfolio Diversification / Application: Measurement of Income Inequality / Generalizations of the Fundamental Theorems of Calculus / The Banach-Zarecki Theorem
Weak Convergence of Probability Measures / Convergence of Random Variables / Poisson Limit Theorem / The Prokhorov Metrization / Properties of P(X) / Prokhorov's Theorem / An Alternative Metrization of P(X) / Application: The Expected Utility Theory
Chapter G: Stochastic Independence
Independence of Classes of Events / Independence of Random Variables / Application: Records / Finite Products of Measure Spaces / n-Dimensional Lebesgue Measure / Tonelli-Fubini Theorems / Application: Markov Processes with the Doeblin Condition / Application: Mixed Equilibria in Games / Infinite Products of Probability Spaces
Chapter H: A Primer on Probability Limit Theorems
Preliminaries / Laws of Large Numbers / The Borel-Cantelli Lemmas / Convergence of Series of Random Variables / Kolmogorov's 0-1 Law / the Hewitt-Savage 0-1 Law / Central Limit Theorems; Trotter's proof
Chapter I: Stationary Sequences and Ergodic Theory
Stationary Random Sequences / Ergodicity / Ergodic Theorems / Applications
Chapter J: Stochastic Dependence
Conditional Expectation / Properties of Conditional Expectation
Chapter K: Martingales
Martingales / Stopped Martingales / The Martingale Convergence Theorems / Applications
Appendix 1: Mathematical Analysis on the Real Line
Appendix 3: Normed Linear Spaces
Elements of Order Theory
Efe A. Ok
Preface (TBW)
Table of Contents
Chapter 1: Preordered Sets and Posets
Binary Relations / Equivalence Relations / Order Relations / Preordered Linear Spaces / Representation through Complete Preorders / Extrema / Parameters of Posets / Suprema and Infima
Elements of Lattice Theory / Modular Lattices / Distributive Lattices / Lattice Ordered Algebraic Systems / Functions on Lattices
Chapter 3: Order-Preserving Maps
Notions of Monotonicity / Order Preservation on Algebraic Structures / Galois Connections / Order-Preserving Correspondences / An Application to Optimization Theory
Chapter 4: Order-Isomorphisms
Order-Isomorphisms / Fundamental Isomorphism Theorems for Lattices / Dedekind-MacNeille Completion
Chapter 5: Mobius Functions
Motivation: Inversion Problems on Posets / Incidence Algebras / Mobius Functions / Mobius Algebras
Interlude: Axiom of Choice
The Axiom of Choice / Digression: Paradoxical Consequences of the Axiom of Choice
Chapter 6: Zorn's Lemma and its Applications
Chains and Antichains, Again / The Hausdorff Maximal Principle / An Application to Optimization Theory / Zorn's Lemma / Applications of Zorn's Lemma / The Well-Ordering Principle
Chapter 6: Order-Theoretic Fixed Point Theory
Fixed Point Theory / Completeness Conditions for Posets, Again / Iterative Fixed Point Theorems / Tarski's Fixed Point Theorems / Converse of the Knaster-Tarski Theorem / The Abian-Brown Fixed Point Theorem / Fixed Points of Order-Preserving Correspondences
Chapter 7: The Brezis-Browder Ordering Principle and its Applications
A Selection of Ordering Principles / Applications to Fixed Point Theory / Applications to Variational Analysis / An Application to Convex Analysis
Chapter 8: Completions and Decompositions of Preordered Sets (TBW)
Chapter 9: Functional (Utility) Representation of Preorders (Incomplete)
Preliminaries / Representation through Order-Separability / Representation through Semicontinuity / The Open Gap Lemma / The Debreu-Eilenberg Representation Theorems / Multi-Utility Representation / Continuous Multi-Utility Representation / Finite Multi-Utility Representation
Chapter 10: Advances in Lattice Theory (TBW)
Appendix: A Primer on Topological Spaces (Incomplete)
Topological Spaces / Metric Spaces / The Hausdorff Metric / Topological Linear Spaces
References
Applied Point-Set Topology
Efe A. Ok
Preface (TBW)
PART 1: Topology of Metric Spaces
Chapter 2: Continuity in Metric Spaces
Chapter 3: Complete Metric Spaces
PART 2: Point-Set Topology
Chapter 5: Products and Quotients
Chapter 8: Topological Dynamics
Chapter 9: Fixed Point Theory
PART 3: More Point-Set Topology
Chapter 12: CW Complexes
PART 4: Topological Algebra
Chapter 13: Topological Groups
Chapter 14: Topological Linear Spaces