## Real Analysis with Economic Applications

### Efe A. Ok

**Book Description and Endorsements**

**Table of Contents**

**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)**

**Table of Contents**

**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)**

**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

**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