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Probabilistic Metric Spaces.

Author: B Schweizer; A Sklar
Publisher: Newburyport : Dover Publications, 2011.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
This distinctly nonclassical treatment focuses on developing aspects that differ from the theory of ordinary metric spaces, working directly with probability distribution functions rather than random variables. The two-part treatment begins with an overview that discusses the theory's historical evolution, followed by a development of related mathematical machinery. The presentation defines all needed concepts,  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Schweizer, B.
Probabilistic Metric Spaces.
Newburyport : Dover Publications, ©2011
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: B Schweizer; A Sklar
ISBN: 9780486143750 0486143759
OCLC Number: 1041841114
Notes: 12.2. Uniform Continuity of the Distance Function.
Description: 1 online resource (569 pages)
Contents: Cover; Title Page; Copyright Page; Contents; Preface to the Dover Edition; Preface; Special Symbols; Chapter 1. Introduction and Historical Survey; 1.0. Introduction; 1.1. Beginnings; 1.2. Menger, 1942; 1.3. Wald, 1943; 1.4. Developments, 1956-1960; 1.5. Some Examples; 1.6. Šerstnev, 1962; 1.7. Random Metric Spaces; 1.8. Topologies; 1.9. Tools; 1.10. Postscript; Chapter 2. Preliminaries; 2.1. Sets and Functions; 2.2. Functions on Intervals; 2.3. Probabilities, Integrals, Random Variables; 2.4. Binary Operations; Chapter 3. Metric and Topological Structures; 3.1. Metric and Related Spaces. 3.2. Isometries, Homotheties, Metric Transforms3.3. Betweenness; 3.4. Minkowski Metrics; 3.5. Topological Structures; Chapter 4. Distribution Functions; 4.1. Spaces of Distribution Functions; 4.2. The Modified Lévy Metric; 4.3. The Space of Distance Distribution Functions; 4.4. Quasi-Inverses of Nondecreasing Functions; Chapter 5. Associativity; 5.1. Associative Binary Operations; 5.2. Generators and Ordinal Sums; 5.3. Associative Functions on Intervals; 5.4. Representation of Archimedean Functions; 5.5. Triangular Norms, Additive and Multiplicative Generators; 5.6. Examples. 5.7. Conorms and Composition Laws5.8. Open Problems; Chapter 6. Copulas; 6.1. Fundamental Properties of n-Increasing Functions; 6.2. Joint Distribution Functions, Subcopulas, and Copulas; 6.3. Copulas and t-Norms; 6.4. Dual Copulas; 6.5. Copulas and Random Variables; 6.6. Margins of Copulas; 6.7. Open Problems; Chapter 7. Triangle Functions; 7.1. Introduction; 7.2. The Operations <U+0078>T, L; 7.3. The Operations <U+0078>T*, L; 7.4. The Operations <U+0076>C, L; 7.5. The Operations <U+0075>C, L; 7.6. Derivability and Nonderivability from Functions of Random Variables; 7.7. Duality; 7.8. The Conjugate Transform. 7.9. Open ProblemsChapter 8. Probabilistic Metric Spaces; 8.1. Probabilistic Metric Spaces in General; 8.2. Transformed Triangle Inequalities and Derived Metrics; 8.3. Equilateral Spaces; 8.4. Simple Spaces; 8.5. Ellipse m-Metrics and Hysteresis; 8.6. <U+00dc>-Simple Spaces; 8.7. Best-Possible Triangle Inequalities; 8.8. Open Problems; Chapter 9. Random Metric Spaces; 9.1. E-Spaces; 9.2. Pseudometrically Generated Spaces, Sherwood's Theorem; 9.3. Random Metric Spaces; 9.4. The Probability of the Triangle Inequality; 9.5. W-Spaces; 9.6. Open Problems; Chapter 10. Distribution-Generated Spaces. 10.1. Introduction10.2. Consistency, Triangle Inequalities; 10.3. C-Spaces; 10.4. Homogeneous and Semihomogeneous C-Spaces; 10.5. Moments and Metrics; 10.6. Normal C-Spaces; 10.7. Moments in Normal C-Spaces; 10.8. Open Problems; Chapter 11. Transformation-Generated Spaces; 11.1. Transformation-Generated Spaces; 11.2. Measure-Preserving Transformations; 11.3. Mixing Transformations; 11.4. Recurrence; 11.5. E-Processes: The Case of Markov Chains; 11.6. Open Problems; Chapter 12. The Strong Topology; 12.1. The Strong Topology and Strong Uniformity.

Abstract:

This distinctly nonclassical treatment focuses on developing aspects that differ from the theory of ordinary metric spaces, working directly with probability distribution functions rather than random variables. The two-part treatment begins with an overview that discusses the theory's historical evolution, followed by a development of related mathematical machinery. The presentation defines all needed concepts, states all necessary results, and provides relevant proofs. The second part opens with definitions of probabilistic metric spaces and proceeds to examinations of special classes of probabilistic metric spaces, topologies, and several related structures, such as probabilistic normed and inner-product spaces. Throughout, the authors focus on developing aspects that differ from the theory of ordinary metric spaces, rather than simply transferring known metric space results to a more general setting.

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