This non-classical treatment focuses on developing aspects that differ from the theory of ordinary metric spaces, working directly with probability distribution functions rather than random variables. Topics include special classes of probabilistic metric spaces, topologies, and several related structures, such as probabilistic normed and inner-product spaces. 1983 edition, updated with 3 new appendixes. Includes 17 illustrations.

Preface to the Dover Edition

vii

Preface

ix

Special Symbols

xiii

Chapter 1. Introduction and Historical Survey

1

(17)

1.0. Introduction

1

(1)

1.1. Beginnings

2

(1)

1.2. Menger, 1942

3

(1)

1.3. Wald, 1943

4

(1)

1.4. Developments, 1956-1960

5

(2)

1.5. Some Examples

7

(2)

1.6. Serstnev, 1962

9

(1)

1.7. Random Metric Spaces

10

(2)

1.8. Topologies

12

(2)

1.9. Tools

14

(3)

1.10. Postscript

17

(1)

Chapter 2. Preliminaries

18

(12)

2.1. Sets and Functions

18

(4)

2.2. Functions on Intervals

22

(2)

2.3. Probabilities, Integrals, Random Variables

24

(3)

2.4. Binary Operations

27

(3)

Chapter 3. Metric and Topological Structures

30

(13)

3.1. Metric and Related Spaces

30

(3)

3.2. Isometries, Homotheties, Metric Transforms

33

(3)

3.3. Betweenness

36

(1)

3.4. Minkowski Metrics

37

(1)

3.5. Topological Structures

38

(5)

Chapter 4. Distribution Functions

43

(11)

4.1. Spaces of Distribution Functions

43

(2)

4.2. The Modified Lévy Metric

45

(3)

4.3. The Space of Distance Distribution Functions

48

(1)

4.4. Quasi-Inverses of Nondecreasing Functions

49

(5)

Chapter 5. Associativity

54

(24)

5.1. Associative Binary Operations

54

(1)

5.2. Generators and Ordinal Sums

55

(2)

5.3. Associative Functions on Intervals

57

(5)

5.4. Representation of Archimedean Functions

62

(3)

5.5. Triangular Norms, Additive and Multiplicative Generators

65

(5)

5.6. Examples

70

(3)

5.7. Conorms and Composition Laws

73

(3)

5.8. Open Problems

76

(2)

Chapter 6. Copulas

78

(18)

6.1. Fundamental Properties of n-Increasing Functions

78

(4)

6.2. Joint Distribution Functions, Subcopulas, and Copulas

82

(3)

6.3. Copulas and t-Norms

85

(3)

6.4. Dual Copulas

88

(1)

6.5. Copulas and Random Variables

89

(3)

6.6. Margins of Copulas

92

(1)

6.7. Open Problems

93

(3)

Chapter 7. Triangle Functions

96

(28)

7.1. Introduction

96

(3)

7.2. The Operations τT,L

99

(5)

7.3. The Operations τT*,L

104

(2)

7.4. The Operations σC,L

106

(3)

7.5. The Operations ρC,L

109

(2)

7.6. Derivability and Nonderivability from Functions of Random Variables

111

(3)

7.7. Duality

114

(3)

7.8. The Conjugate Transform

117

(3)

7.9. Open Problems

120

(4)

Chapter 8. Probabilistic Metric Spaces

124

(18)

8.1. Probabilistic Metric Spaces in General

124

(3)

8.2. Transformed Triangle Inequalities and Derived Metrics

127

(4)

8.3. Equilateral Spaces

131

(1)

8.4. Simple Spaces

132

(1)

8.5. Ellipse m-Metrics and Hysteresis

133

(5)

8.6. α-Simple Spaces

138

(2)

8.7. Best-Possible Triangle Inequalities

140

(1)

8.8. Open Problems

141

(1)

Chapter 9. Random Metric Spaces

142

(15)

9.1. E-Spaces

142

(3)

9.2. Pseudometrically Generated Spaces, Sherwood's Theorem

145

(4)

9.3. Random Metric Spaces

149

(3)

9.4. The Probability of the Triangle Inequality

152

(3)

9.5. W-Spaces

155

(1)

9.6. Open Problems

156

(1)

Chapter 10. Distribution-Generated Spaces

157

(18)

10.1. Introduction

157

(2)

10.2. Consistency, Triangle Inequalities

159

(2)

10.3. C-Spaces

161

(4)

10.4. Homogeneous and Semihomogeneous C-Spaces

165

(3)

10.5. Moments and Metrics

168

(3)

10.6. Normal C-Spaces

171

(1)

10.7. Moments in Normal C-Spaces

172

(2)

10.8. Open Problems

174

(1)

Chapter 11. Transformation-Generated Spaces

175

(16)

11.1. Transformation-Generated Spaces

175

(3)

11.2. Measure-Preserving Transformations

178

(3)

11.3. Mixing Transformations

181

(2)

11.4. Recurrence

183

(3)

11.5. E-Processes: The Case of Markov Chains

186

(3)

11.6. Open Problems

189

(2)

Chapter 12. The Strong Topology

191

(28)

12.1. The Strong Topology and Strong Uniformity

191

(4)

12.2. Uniform Continuity of the Distance Function

195

(3)

12.3. Examples

198

(2)

12.4. The Probabilistic Diameter

200

(2)

12.5. Completion of Probabilistic Metric Spaces

202

(1)

12.6. Contraction Maps

203

(5)

12.7. Product Spaces

208

(4)

12.8. Countable Products

212

(2)

12.9. The Probabilistic Hausdorff Distance

214

(1)

12.10. Discernibility Relations

215

(2)

12.11. Open Problems

217

(2)

Chapter 13. Profile Functions

219

(7)

13.1. Profile Closures

219

(3)

13.2. Distinguishability

222

(4)

Chapter 14. Betweenness

226

(10)

14.1. Wald Betweenness

226

(4)

14.2. Transform Betweenness

230

(1)

14.3. Menger Betweenness

231

(2)

14.4. Probabilistic Betweenness

233

(2)

14.5. Open Problems

235

(1)

Chapter 15. Supplements

236

(17)

15.1. Probabilistic Normed Spaces

236

(4)

15.2. Probabilistic Inner Product Spaces

240

(2)

15.3. Probabilistic Topologies

242

(4)

15.4. Probabilistic Information Spaces

246

(3)

15.5. Generalized Metric Spaces

249

(4)

References

253

(14)

Index

267

(10)

Errata

277

(4)

Notes

281

(24)

Supplementary References

305

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PROBABILISTIC METRIC SPACES

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