Measure and Integral - 1st Edition - ISBN: 9780123785503, 9781483263045

Measure and Integral

1st Edition

Authors: Konrad Jacobs
Editors: Z. W. Birnbaum E. Lukacs
eBook ISBN: 9781483263045
Imprint: Academic Press
Published Date: 28th November 1978
Page Count: 592
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Description

Probability and Mathematical Statistics: Measure and Integral provides information pertinent to the general mathematical notions and notations. This book discusses how the machinery of ?-extension works and how ?-content is derived from ?-measure.

Organized into 16 chapters, this book begins with an overview of the classical Hahn–Banach theorem and introduces the Banach limits in the form of a major exercise. This text then presents the Daniell extension theory for positive ?-measures. Other chapters consider the transform of ?-contents and ?-measures by measurable mappings and kernels. This text is also devoted to a thorough study of the vector lattice of signed contents. This book discusses as well an abstract regularity theory and applied to the standard cases of compact, locally compact, and Polish spaces. The final chapter deals with the rudiments of the Krein–Milman theorem, along with some of their applications.

This book is a valuable resource for graduate students.

Table of Contents


Preface

Chapter 0 Basic Notions and Notations

1. Sets, Relations, and Mappings

2. Vector Spaces and Vector Lattices. Banach Spaces

3. Topological Spaces

Chapter 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 Spaces

Chapter 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. Completion

Chapter 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 Integrability

Chapter IV Transform of σ-Contents

1. Measurable Mappings

2. Transform of σ-Additive Functions

3. Ergodic Theorems

4. Kernels

Chapter 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 Spaces

Chapter 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 Structure

Chapter 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 Variation

Chapter 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 Problem

Chapter 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 Measures

Chapter 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 Spaces

Chapter 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 Groups

Chapter 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 Theorem

Chapter XIV Atoms, Conditional Atoms, and Entropy

1. Atoms and Conditional Atoms

2. Entropy

Chapter XV Convex Compact Sets and Their Extremal Points

1. Locally Convex Topological Vector Spaces

2. Barycenters

3. Applications

Chapter XVI Lifting

1. The Notion and Existence of a Lifting

2. Strong Lifting

3. Applications

Appendix 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-Integral

Appendix 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 Topology

Selected Bibliography

Index

Details

No. of pages:
592
Language:
English
Copyright:
© Academic Press 1978
Published:
Imprint:
Academic Press
eBook ISBN:
9781483263045

About the Author

Konrad Jacobs

About the Editor

Z. W. Birnbaum

E. Lukacs

Affiliations and Expertise

Bowling Green State University