Vector Measures

Vector Measures

1st Edition - January 1, 1967

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  • Editors: I. N. Sneddon, M. Stark
  • eBook ISBN: 9781483222653

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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.

Table of Contents

  • 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 𝓒(𝓐)

    § 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 Measure

    Chapter 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(𝒞)

    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 ℳ∞E and ℒ∞E

    1. Integration of Totally Measurable Functions

    2. Linear Operations on the Space ℳE(𝒞)

    3. Dominated Operations

    4. The Semi-Norm N∞

    5. Almost Totally Measurable Functions

    6. Operations on ℳ∞E(𝒞)

    7. The Space ℒ∞E(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 ℒ∞

    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. Integral Representation of Linear Operations on ℒpE

    Chapter III. Regular Measures

    § 14. Borel Sets. Borei Measures

    1. The Semi-Tribe of the Relatively Compact Borel Sets

    2. The Tribe of the Borel Sets

    3. Baire Sets

    4. Borel Measures. Baire Measures

    5. Integration with Respect to a Borel Measure

    § 15. Regular Measures

    1. Topologisation of 𝓟(T)

    2. Regular Set Functions

    3. Regular Additive Set Functions on a Clan

    4. Set Functions Regular on a Subclass

    5. Regular Positive Set Functions

    6. Regular Set Functions with Finite Variation

    7. Integration with Respect to Regular Measures

    8. Measurable Functions

    § 16. Construction and Extension of Regular Measures

    1. Extension of Regular Set Functions

    2. Construction of Regular Positive Borel Measures

    3. Construction of Regular Borel Vector Measures

    4. Regularity of the Vector Measures on a Subclass

    § 17. Stieltjes Measures on the Real Line

    1. Functions with Finite Variation

    2. Stieltjes Measures Deduced from Functions with Finite Variation

    3. Functions with Finite Variation Deduced from Stieltjes Measures

    § 18. Reduction of the Integration on Abstract Spaces to the Integration on Locally Compact Spaces

    1. Theorem of Stone

    2. Theorem of Kakutani

    § 19. Integral Representation of Linear Operations on 𝓚E(T)

    1. The Space 𝓚E(T)

    2. Linear Operations on 𝓚E(T)

    3. Dominated Operations

    4. Positive Functional on 𝓚(T)

    5. Continuous Linear Operations

    § 20. Disintegration of Measures

    1. Images of Measures

    2. The Strong Lifting Property

    3. Disintegration of Measures

    Notes and Remarks


    Index of Notations


    Other Titles Published in this Series

Product details

  • No. of pages: 446
  • Language: English
  • Copyright: © Pergamon 1967
  • Published: January 1, 1967
  • Imprint: Pergamon
  • eBook ISBN: 9781483222653

About the Editors

I. N. Sneddon

M. Stark

About the Author

N. Dinculeanu

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