Measure and Integral

PrefaceChapter 0 Basic Notions and Notations 1. Sets, Relations, and Mappings 2. Vector Spaces and Vector Lattices. Banach Spaces 3. Topological SpacesChapter I Positive Contents and Measures 1. Contents on Set Rings 2. σ-Contents 3. Eudoxos Extension of Contents 4. σ-Rings, Local σ-Rings, and σ-Fields 5. Uniqueness of Extension of σ-Contents 6. The Hahn-Banach Theorem 7. Elementary Domains 8. Measures on Elementary Domains 9. Riemann (Eudoxos) Extension of Positive Measures 10. Measure Spaces over Topological Spaces 11. The Extension Problem for Measure SpacesChapter II Extension of σ-Contents After Carathéodory 1. The Outer Content Derived from a Given Content 2. Additive Decomposers 3. Routine Extension from Local σ-Rings to σ-Rings 4. Minimal Extension to a σ-Field 5. Definition of a σ-Content from Local Data 6. CompletionChapter III Extension of Positive σ- and σ-Measures, After Daniell 1. Extension Step I: σ-Upper and σ-Lower Functions, Monotone Extension 2. Extension Step II: Squeezing-in and the Definition of Integrable Functions 3. The Upper and the Lower Integral 4. Nullfunctions and Nullsets 5. Basic Theorems for the Integral 6. The σ-Content Derived from a σ-Measure 7. τ-Extension of a τ-Measure 8. Measurability of Real and Complex Functions 9. Measurability and Integrability 10. Integration of Complex-Valued Functions 11. The Real and the Complex Hubert Space L2 12. Stochastic Convergence and Uniform IntegrabilityChapter IV Transform of σ-Contents 1. Measurable Mappings 2. Transform of σ-Additive Functions 3. Ergodic Theorems 4. KernelsChapter V Contents and Measures in Topological Spaces. Part I: Regularity 1. The General Concept of Regularity 2. Regularity of σ-Contents in Topological Spaces 3. Regularity of σ-Contents in Compact Spaces 4. Regularity of σ-Contents in Locally Compact Spaces 5. Regularity in Polish SpacesChapter VI Contents and Measures in Product Spaces 1. Set Systems in Product Spaces 2. Two Factors 3. Finitely Many Factors 4. Countably Many Factors 5. Arbitrarily Many Factors 6. Independence 7. Markovian Semigroups and Their Path StructureChapter VII Set Functions in General 1. Basic Notions for Set Functions 2. The Additive and σ-Additive Parts of a Superadditive Set Function 3. σ-Additivity. The Vitali-Hahn-Saks Theorem 4. Total VariationChapter VIII The Vector Lattice of Signed Contents 1. Signed Contents and Signed σ-Contents 2. Hahn Decompositions 3. Absolute Continuity of Signed σ-Contents 4. Lebesgue Decompositions 5. The Radon-Nikodym Theorem for Signed σ-Contents with Finite Total Variation 6. Conditional Expectations 7. Martingales, Submartingales, and Supermartingales 8. The Radon-Nikodym ProblemChapter IX The Vector Lattice of Signed Measures 1. Abstract Vector Lattices and Their Duals 2. Signed Measures 3. Absolute Continuity of Measures 4. Signed Contents and Signed MeasuresChapter X The Spaces Lp 1. The Spaces Lpm (1 ≤ p ≤ χ) 2. Duality of the Spaces Lpm (1 ≤ p ≤ χ)Chapter XI Contents and Measures in Topological Spaces. Part II: The Weak Topology 1. The Weak Topology for σ-Contents in Arbitrary Topological Spaces 2. The Weak Topology for σ-Contents in Polish SpacesChapter XII The Haar Measure on Locally Compact Groups 1. The Haar Measure on Compact Groups 2. Definition and Basic Machinery of Locally Compact Groups 3. The Haar Measure on Locally Compact GroupsChapter XIII Souslin Sets, Analytic Sets, and Capacities 1. Souslin Extensions 2. Souslin Sets and Analytic Sets in Polish Spaces 3. Capacities 4. The Capacity Approach to σ-Content Extension 5. The Measurable Choice TheoremChapter XIV Atoms, Conditional Atoms, and Entropy 1. Atoms and Conditional Atoms 2. EntropyChapter XV Convex Compact Sets and Their Extremal Points 1. Locally Convex Topological Vector Spaces 2. Barycenters 3. ApplicationsChapter XVI Lifting 1. The Notion and Existence of a Lifting 2. Strong Lifting 3. ApplicationsAppendix A The Perron-Ward Integral and Related Concepts 1. Notations 2. The S-Integral 3. The V-Integral 4. The Perron-Ward Integral 5. Monotone Convergence 6. Relation to the Daniell Integral 7. Some Results on the S-IntegralAppendix B Contents with Given Marginals 1. The Ford-Fulkerson Theorem 2. Matrices with Given Marginals 3. Contents and σ-Contents with Given Marginals 4. Results Involving TopologySelected BibliographyIndex