An Introduction to Measure-Theoretic Probability

Pictured on the Cover

• Carathéodory, Constantine (1873–1950)

Preface to First Edition

Preface to Second Edition

Chapter 1. Certain Classes of Sets, Measurability, and Pointwise Approximation

• Abstract
• 1.1 Measurable Spaces
• 1.2 Product Measurable Spaces
• 1.3 Measurable Functions and Random Variables

Chapter 2. Definition and Construction of a Measure and its Basic Properties

• Abstract
• 2.1 About Measures in General, and Probability Measures in Particular
• 2.2 Outer Measures
• 2.3 The Carathéodory Extension Theorem
• 2.4 Measures and (Point) Functions

Chapter 3. Some Modes of Convergence of Sequences of Random Variables and their Relationships

• Abstract
• 3.1 Almost Everywhere Convergence and Convergence in Measure
• 3.2 Convergence in Measure is Equivalent to Mutual Convergence in Measure

Chapter 4. The Integral of a Random Variable and its Basic Properties

• Abstract
• 4.1 Definition of the Integral
• 4.2 Basic Properties of the Integral
• 4.3 Probability Distributions

Chapter 5. Standard Convergence Theorems, The Fubini Theorem

• Abstract
• 5.1 Standard Convergence Theorems and Some of Their Ramifications
• 5.2 Sections, Product Measure Theorem, the Fubini Theorem

Chapter 6. Standard Moment and Probability Inequalities, Convergence in the rth Mean and its Implications

• Abstract
• 6.1 Moment and Probability Inequalities
• 6.2 Convergence in the rth Mean, Uniform Continuity, Uniform Integrability, and their Relationships

Chapter 7. The Hahn–Jordan Decomposition Theorem, The Lebesgue Decomposition Theorem, and the Radon–Nikodym Theorem

• Abstract
• 7.1 The Hahn–Jordan Decomposition Theorem
• 7.2 The Lebesgue Decomposition Theorem
• 7.3 The Radon–Nikodym Theorem

Chapter 8. Distribution Functions and Their Basic Properties, Helly–Bray Type Results

• Abstract
• 8.1 Basic Properties of Distribution Functions
• 8.2 Weak Convergence and Compactness of a Sequence of Distribution Functions
• 8.3 Helly–Bray Type Theorems for Distribution Functions

Chapter 9. Conditional Expectation and Conditional Probability, and Related Properties and Results

• Abstract
• 9.1 Definition of Conditional Expectation and Conditional Probability
• 9.2 Some Basic Theorems About Conditional Expectations and Conditional Probabilities
• 9.3 Convergence Theorems and Inequalities for Conditional Expectations
• 9.4 Further Properties of Conditional Expectations and Conditional Probabilities

Chapter 10. Independence

• Abstract
• 10.1 Independence of Events, σ-Fields, and Random Variables
• 10.2 Some Auxiliary Results
• 10.3 Proof of Theorem 1 and of Lemma 1 in Chapter 9

Chapter 11. Topics from the Theory of Characteristic Functions

• Abstract
• 11.1 Definition of the Characteristic Function of a Distribution and Basic Properties
• 11.2 The Inversion Formula
• 11.3 Convergence in Distribution and Convergence of Characteristic Functions—The Paul Lévy Continuity Theorem
• 11.4 Convergence in Distribution in the Multidimensional Case—The Cramér–Wold Device
• 11.5 Convolution of Distribution Functions and Related Results
• 11.6 Some Further Properties of Characteristic Functions
• 11.7 Applications to the Weak Law of Large Numbers and the Central Limit Theorem
• 11.8 The Moments of a Random Variable Determine its Distribution
• 11.9 Some Basic Concepts and Results from Complex Analysis Employed in the Proof of Theorem 11

Chapter 12. The Central Limit Problem: The Centered Case

• Abstract
• 12.1 Convergence to the Normal Law (Central Limit Theorem, CLT)
• 12.2 Limiting Laws of L(Sn) Under Conditions (C)
• 12.3 Conditions for the Central Limit Theorem to Hold
• 12.4 Proof of Results in Section 12.2

Chapter 13. The Central Limit Problem: The Noncentered Case

• Abstract
• 13.1 Notation and Preliminary Discussion
• 13.2 Limiting Laws of L(Sn) Under Conditions (C″)
• 13.3 Two Special Cases of the Limiting Laws of L(Sn)

Chapter 14. Topics from Sequences of Independent Random Variables

• Abstract
• 14.1 Kolmogorov Inequalities
• 14.2 More Important Results Toward Proving the Strong Law of Large Numbers
• 14.3 Statement and Proof of the Strong Law of Large Numbers
• 14.4 A Version of the Strong Law of Large Numbers for Random Variables with Infinite Expectation
• 14.5 Some Further Results on Sequences of Independent Random Variables

Chapter 15. Topics from Ergodic Theory

• Abstract
• 15.1 Stochastic Process, the Coordinate Process, Stationary Process, and Related Results
• 15.2 Measure-Preserving Transformations, the Shift Transformation, and Related Results
• 15.3 Invariant and Almost Sure Invariant Sets Relative to a Transformation, and Related Results
• 15.4 Measure-Preserving Ergodic Transformations, Invariant Random Variables Relative to a Transformation, and Related Results
• 15.5 The Ergodic Theorem, Preliminary Results
• 15.6 Invariant Sets and Random Variables Relative to a Process, Formulation of the Ergodic Theorem in Terms of Stationary Processes, Ergodic Processes

Chapter 16. Two Cases of Statistical Inference: Estimation of a Real-Valued Parameter, Nonparametric Estimation of a Probability Density Function

• Abstract
• 16.1 Construction of an Estimate of a Real-Valued Parameter
• 16.2 Construction of a Strongly Consistent Estimate of a Real-Valued Parameter
• 16.3 Some Preliminary Results
• 16.4 Asymptotic Normality of the Strongly Consistent Estimate
• 16.5 Nonparametric Estimation of a Probability Density Function
• 16.6 Proof of Theorems 3–5

Appendix A. Brief Review of Chapters 1–16

• Chapter 1 Certain Classes of Sets, Measurability, and Pointwise Approximation
• Chapter 2 Definition and Construction of a Measure and its Basic Properties
• Chapter 3 Some Modes of Convergence of Sequences of Random Variables and their Relationships
• Chapter 4 The Integral of a Random Variable and its Basic Properties
• Chapter 5 Standard Convergence Theorems, The Fubini Theorem
• Chapter 6 Standard Moment and Probability Inequalities, Convergence in the rth Mean and its Implications
• Chapter 7 The Hahn–Jordan Decomposition Theorem, The Lebesgue Decomposition Theorem, and the Radon–Nikodym Theorem
• Chapter 8 Distribution Functions and Their Basic Properties, Helly–Bray Type Results
• Chapter 9 Conditional Expectation and Conditional Probability, and Related Properties and Results
• Chapter 10 Independence
• Chapter 11 Topics from the Theory of Characteristic Functions
• Chapter 12 The Central Limit Problem: The Centered Case
• Chapter 13 The Central Limit Problem: The Non-centered Case
• Chapter 14 Topics from Sequences of Independent Random Variables
• Chapter 15 Topics from Ergodic Theory
• Chapter 16 Two Cases of Statistical Inference: Estimation of a Real-valued Parameter, Nonparametric Estimation of a Probability Density Function

Appendix B. Brief Review of Riemann–Stieltjes Integral

Appendix C. Notation and Abbreviations

Selected References

Revised Answers Manual to an Introduction to Measure-Theoretic Probability

• Chapter 1
• Chapter 2
• Chapter 3
• Chapter 4
• Chapter 5
• Chapter 6
• Chapter 7
• Chapter 8
• Chapter 9
• Chapter 10
• Chapter 11
• Chapter 12
• Chapter 13
• Chapter 14
• Chapter 15