Probability Algebras and Stochastic Spaces

Probability Algebras and Stochastic Spaces explores the fundamental notions of probability theory in the so-called “point-free” way. The space of all elementary random variables defined over a probability algebra in a “point-free” way is a base for the stochastic space of all random variables, which can be obtained from it by lattice-theoretic extension processes.

This book is composed of eight chapters and begins with discussions of the definition, properties, scope, and extension of probability algebras. The succeeding chapters deal with the Cartesian product of probability algebras and the principles of stochastic spaces. These topics are followed by surveys of the expectation, moments, and spaces of random variables. The final chapters define generalized random variables and the Boolean homomorphisms of these variables.

This book will be of great value to mathematicians and advance mathematics students.

Preface

Chapter I—Probability Algebras

1. Definition and Properties

2. Probability Subalgebras

3. Isometric Probability Algebras

4. Examples

5. Separability Relative to a Probability

6. Countably Additive (σ-Additive) Probabilities

7. Probability σ-Algebras

8. Quasi-Probability Algebras

9. Probability Spaces

Chapter II—Extension of Probability Algebras

1. The Probability Algebra as a Metric Space

2. Construction of a σ-Extension of a Probability Algebra

3. The Linear Lebesgue Probability σ-Algebra

4. Classification of the p-Separable Probability σ-Algebras

Chapter III—Cartesian Product of Probability Algebras

1. Cartesian Product of Boolean Alegbras

2. Product Probability Algebras

3. Classification of Probability σ-Algebras

4. Representation of Probability σ-Algebras by Probability Spaces

5. Independence in Probability

Chapter IV—Stochastic Spaces

1. Experiments (Trials) in Probability Algebras

2. Elementary Random Variables (Elementary Stochastic Space)

3. Convergence in Stochastic Spaces

4. o-Convergence in ℰ with Respect to a Vector Sublattice of ℰ

5. Extension of the Elementary Stochastic Space

6. Stochastic Space of All Bounded Random Variables

7. Convergence in Probability and Almost Uniform Convergence

8. Generators of the Stochastic Space

9. Other Convergences in the Stochastic Space

10. Closure Operator in the Stochastic Space

11. Series Convergence

12. Composition of Random Variables

Chapter V—Expectation of Random Variables

1. Expectation of Elementary Random Variables

2. The Space ℒ1 of All rv's with Expectation

3. Signed Measures

4. The Radon-Nikodym Theorem

5. Remarks

Chapter VI—Moments. Spaces ℒq

1. Powers of rv's

2. Moments of Random Variables

3. The Spaces ℒq

4. Convergence in Mean and Equi-Integrability

Chapter VII—Generalized Random Variables (Random Variables Having Values in Any Space)

1. Preliminaries

2. Generalized Elementary Random Variables

3. Completion with Respect to o-Convergence

4. Completion with Respect to a Norm

5. Expectation of rv's Having Values in a Banach Space

6. The Spaces Lq of rv's Having Values in a Banach Space. Moments

Chapter VIII—Complements

1. The Radon-Nikodym Theorem For the Bochner Integral

2. Conditional Probability

3. Conditional Expectation

4. Distributions of Random Variables

5. Boolean Homomorphisms of Random Variables

Appendix I—Lattices

1. Partially Ordered Sets

2. Lattices

3. Boolean Algebras

4. Homomorphisms and Ideals of a Boolean Algebra

5. Order Convergence

6. Closures

Appendix II—Lattice Groups, Vector Lattices

1. Lattice Groups

2. Vector Lattices or Linear Lattice Spaces

Bibliographical Notes

Bibliography

List of Symbols

Index