## Real Analysis with Economic Applications

### Efe A. Ok

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 with Economic Applications

### Efe A. Ok

Preface (TBW)

Chapter A: Preliminaries
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

Chapter C: Random Variables
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 / Spaces of Integrable Random Variables / The Riesz-Radon Representation Theorem / Choquet's Theorem

Chapter E: Expectation via the Stieltjes Integral
The Stieltjes Integral / The Riemann Integral / Absolute Continuity, Again / (Generalized) Fundamental Theorems of Calculus / The Banach-Zarecki Theorem / Expectation as a Stieltjes Integral / Integration by Parts / Application: More on Stochastic Dominance / Economic Applications of Stochastic Dominance Theory

Chapter F: Weak Convergence
Weak Convergence of Probability Measures / Convergence of Random Variables / The Prokhorov Metrization / Properties of P(X) / An Alternative Metrization of P(X)

Chapter G: Applications to Decision-Making under Risk and Uncertainty
The Expected Utility Theorem / Decision-Making Under Uncertainty

Chapter H: Stochastic Independence
Independence of Classes of Events / Independence of Random Variables / Finite Products of Probability Spaces / Application: Nash Equilibrium in Mixed Strategies / Infinite Products of Probability Spaces

Chapter I: 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

Chapter J: Stationary Sequences and Ergodic Theory
Stationary Random Sequences / Ergodicity / Ergodic Theorems / Applications

Chapter K: Conditional Expectation
Conditional Expectation / Properties of Conditional Expectation

Chapter L: Martingales
Martingales / Stopped Martingales / The Martingale Convergence Theorems / Applications

Appendix 1: Mathematical Analysis on the Real Line

Appendix 2: Metric Spaces

Appendix 3: Normed Linear Spaces

## Elements of Order Theory

### Efe A. Ok

Preface (TBW)

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

Chapter 2: Lattices
Elements of Lattice Theory / Modular Lattices / Distributive Lattices / Functions on Lattices

Chapter 3: Order-Preserving Maps and Isomorphisms
Order-Preserving Maps / Fundamental Isomorphism Theorems for Lattices / Order-Preservation on Vector Lattices / Galois Connections / Order-Preserving Correspondences / An Application to Optimization Theory

Chapter 4: 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 5: 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

## Applied Topology

### Preface (TBW)

Chapter 1: Metric Spaces

Chapter 2: Continuity in Metric Spaces

Chapter 3: Complete Metric Spaces

Chapter 4: Topological Spaces

Chapter 5: Products and Quotients

Chapter 6: Connectedness

Chapter 7: Compactness

Chapter 8: Normal Spaces

Chapter 9: Paracompactness

Chapter 10: Topological Algebra

Chapter 11: Fixed Point Theory