Probability Algebras and Stochastic Spaces

PrefaceChapter 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 SpacesChapter 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-AlgebrasChapter 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 ProbabilityChapter 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 VariablesChapter 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. RemarksChapter VI—Moments. Spaces ℒq 1. Powers of rv's 2. Moments of Random Variables 3. The Spaces ℒq 4. Convergence in Mean and Equi-IntegrabilityChapter 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. MomentsChapter 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 VariablesAppendix I—Lattices 1. Partially Ordered Sets 2. Lattices 3. Boolean Algebras 4. Homomorphisms and Ideals of a Boolean Algebra 5. Order Convergence 6. ClosuresAppendix II—Lattice Groups, Vector Lattices 1. Lattice Groups 2. Vector Lattices or Linear Lattice SpacesBibliographical NotesBibliographyList of SymbolsIndex