Beschreibung
Inhaltsangabe0. Conventions and Notation.- 1. Notation: Euclidean space.- 2. Operations involving ±?.- 3. Inequalities and inclusions.- 4. A space and its subsets.- 5. Notation: generation of classes of sets.- 6. Product sets.- 7. Dot notation for an index set.- 8. Notation: sets defined by conditions on functions.- 9. Notation: open and closed sets.- 10. Limit of a function at a point.- 11. Metric spaces.- 12. Standard metric space theorems.- 13. Pseudometric spaces.- I. Operations on Sets.- 1. Unions and intersections.- 2. The symmetric difference operator ?.- 3. Limit operations on set sequences.- 4. Probabilistic interpretation of sets and operations on them.- II. Classes of Subsets of a Space.- 1. Set algebras.- 2. Examples.- 3. The generation of set algebras.- 4. The Borel sets of a metric space.- 5. Products of set algebras.- 6. Monotone classes of sets.- III. Set Functions.- 1. Set function definitions.- 2. Extension of a finitely additive set function.- 3. Products of set functions.- 4. Heuristics on a algebras and integration.- 5. Measures and integrals on a countable space.- 6. Independence and conditional probability (preliminary discussion).- 7. Dependence examples.- 8. Inferior and superior limits of sequences of measurable sets.- 9. Mathematical counterparts of coin tossing.- 10. Setwise convergence of measure sequences.- 11. Outer measure.- 12. Outer measures of countable subsets of R.- 13. Distance on a set algebra defined by a subadditive set function.- 14. The pseudometric space defined by an outer measure.- 15. Nonadditive set functions.- IV. Measure Spaces.- 1. Completion of a measure space (S, S,?).- 2. Generalization of length on R.- 3. A general extension problem.- 4. Extension of a measure defined on a set algebra.- 5. Application to Borel measures.- 6. Strengthening of Theorem 5 when the metric space S is complete and separable.- 7. Continuity properties of monotone functions.- 8. The correspondence between monotone increasing functions on R and measures on B(R).- 9. Discrete and continuous distributions on R.- 10. Lebesgue-Stieltjes measures on RN and their corresponding monotone functions.- 11. Product measures.- 12. Examples of measures on RN.- 13. Marginal measures.- 14. Coin tossing.- 15. The Carathéodory measurability criterion.- 16. Measure hulls.- V. Measurable Functions.- 1. Function measurability.- 2. Function measurability properties.- 3. Measurability and sequential convergence.- 4. Baire functions.- 5. Joint distributions.- 6. Measures on function (coordinate) space.- 7. Applications of coordinate space measures.- 8. Mutually independent random variables on a probability space.- 9. Application of independence: the 0-1 law.- 10. Applications of the 0-1 law.- 11. A pseudometric for real valued measurable functions on a measure space.- 12. Convergence in measure.- 13. Convergence in measure vs. almost everywhere convergence.- 14. Almost everywhere convergence vs. uniform convergence.- 15. Function measurability vs. continuity.- 16. Measurable functions as approximated by continuous functions.- 17. Essential supremum and infimum of a measurable function.- 18. Essential supremum and infimum of a collection of measurable functions.- VI. Integration.- 1. The integral of a positive step function on a measure space (S, S,?,).- 2. The integral of a positive function.- 3. Integration to the limit for monotone increasing sequences of positive functions.- 4. Final definition of the integral.- 5. An elementary application of integration.- 6. Set functions defined by integrals.- 7. Uniform integrability test functions.- 8. Integration to the limit for positive integrands.- 9. The dominated convergence theorem.- 10. Integration over product measures.- 11. Jensen's inequality.- 12. Conjugate spaces and Hölder's inequality.- 13. Minkowski's inequality.- 14. The LP spaces as normed linear spaces.- 15. Approximation of LP functions.- 16. Uniform integrability.- 17. Uniform integrability in terms of uniform integrability
