Vector Measures
Published on 21. July 2014
446 pages
978-1-4832-2265-3 (ISBN)
Description
International Series of Monographs in Pure and Applied Mathematics, Volume 95: Vector Measures focuses on the study of measures with values in a Banach space, including positive measures with finite or infinite values. This book is organized into three chapters. Chapter I covers classes of sets, set functions, variation and semi-variation of set functions, and extension of set functions from a certain class to a wider one. The integration of vector functions with respect to vector measures is reviewed in Chapter II. In Chapter III, the regular measures on a locally compact space and integral representation of the dominated operations on the space of continuous functions with compact carrier are described. This volume is intended for specialists, researchers, and students interested in vector measures.
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Content
Chapter I. Vector Measures § 1. Classes of Sets 1. Clans 2. Tribes 3. Semi-Tribes 4. Semi-Clans and Lattices 5. Monotone Classes 6. The Classes ??(??) § 2. Set Functions 1. Additive Set Functions 2. Positive Additive Set Functions on a Clan 3. Countably Additive Set Functions 4. Measures 5. Positive Measures 6. Operator Set Functions 7. Complex Set Functions 8. Uniqueness of the Set Functions 9. Atomic Measures § 3. Variation of Set Functions 1. Definition of the Variation 2. Properties of the Variation 3. Variation of the Scalar Additive Set Functions 4. Set Functions with Finite Variation 5. Locally Bounded Set Functions 6. Variation of the Scalar Measures on a Semi-Tribe § 4. Semi-Variation of Set Functions 1. Definition of the Semi-Variation 2. Properties of The Semi-Variation 3. Semi-Variation of Set Functions with Values in a Conjugate Space 4. Set Functions with Finite Semi-Variation § 5. Extension of Set Functions 1. Extension of Additive Set Functions 2. Completion of an Additive Set Function 3. Extension of Set Functions with Finite Variation 4. Extension of Positive Measures 5. Jordan MeasureChapter II. Integration § 6. Measurable Functions 1. Step Functions 2. Totally Measurable Functions 3. Real Functions Measurable with Respect to a Tribe 4. Sequences of Measurable Real Functions 5. Measurable Functions with Respect to a Measure 6. Sequences of µ-Measurable Functions 7. Simply Measurable Operator Functions 8. Weakly Measurable Operator Functions § 7. Integration of Step Functions 1. Definition and Properties 2. Convergence in Mean on the Space EE(??) 3. Cauchy Sequences of Step Functions § 8. Integrable Functions 1. Definition and Properties 2. Convergence in Mean on the Space L1E 3. Criteria of Integrability 4. The Space L1 § 9. The Spaces M8E and L8E 1. Integration of Totally Measurable Functions 2. Linear Operations on the Space ME(??) 3. Dominated Operations 4. The Semi-Norm N8 5. Almost Totally Measurable Functions 6. Operations on M8E(??) 7. The Space L8E(µ) § 10. Measures Defined by Densities 1. Locally Integrable Functions 2. Measures Defined by Densities 3. Integration with Respect to a Positive Measure Defined by Density 4. Integration with Respect to a Vector Measure Defined by Density 5. Properties of Measures Defined by Densities 6. Absolutely Continuous Measures 7. Measures with the Direct Sum Property 8. The Theorem of Lebesgue-Nikodym 9. Further Properties of the Measures Defined by Densities 10. Singular Measures 11. Conditional Expectations 12. Martingales 13. Convergence Theorems for Martingales § 11. The Lifting Property of the Space L8 1. Definition and Properties 2. Lifting on Sets 3. Linear Liftings 4. The Existence of the Lifting 5. Limits of Measurable Functions 6. Functions with the Lifting Property § 12. The Spaces LpE 1. Definition and Properties 2. The Inequalities of Holder and Minkowski 3. Convergence in Mean of Order p 4. Computation of the Semi-Norm Np 5. Relations Between the Spaces LpE § 13. Linear Operations on LpE 1. The q-Variation 2. The q-Semi-Variation 3. Linear Operations on LpE 4. The Generalized Theorem of Lebesgue-Nikodym 5.
