Probabilistic metric spaces

383 Pages·2011·9.459 MB·English

by  Schweizer, B., Sklar, A

PROBABILISTIC
METRIC SPACES
B. SCHWEIZER
Professor Emeritus of Mathematics
University of Massachusetts
A. SKLAR
Professor Emeritus of Mathematics
Illinois Institute of Technology
DOVER PUBLICATIONS, INC.
Mineola, New York
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PROBABILISTIC
METRIC SPACES
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Copyright
Copyright © 1983 by Elsevier Science Publishing Co., Inc.
Copyright © 2005 by B. Schweizer and A. Sklar
All rights reserved.
Bibliographical Note
This Dover edition, first published in 2005, is an unabridged republication of the
work first published by Elsevier Science Publishing Co., Inc., in 1983. A new Preface to
the Dover edition, list of Errata, Notes, and Supplementary References have been added.
International Standard Book Number: 978-0-486-14375-0
Manufactured in the United States by Courier Corporation
44514302
www.doverpublications.com
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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 Transforms
3.3. 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
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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 Laws
5.8. 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 τ
T, L
7.3. The Operations τ
T*, L
7.4. The Operations σ
C, L
7.5. The Operations ρ
C, L
7.6. Derivability and Nonderivability from Functions of
Random Variables
7.7. Duality
7.8. The Conjugate Transform
7.9. Open Problems
Chapter 8. Probabilistic Metric Spaces
8.1. Probabilistic Metric Spaces in General
8.2. Transformed Triangle Inequalities and Derived Metrics
8.3. Equilateral Spaces
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8.4. Simple Spaces
8.5. Ellipse m-Metrics and Hysteresis
8.6. α-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. Introduction
10.2. 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
12.2. Uniform Continuity of the Distance Function
12.3. Examples
12.4. The Probabilistic Diameter
12.5. Completion of Probabilistic Metric Spaces
12.6. Contraction Maps
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12.7. Product Spaces
12.8. Countable Products
12.9. The Probabilistic Hausdorff Distance
12.10. Discemibility Relations
12.11. Open Problems
Chapter 13. Profile Functions
13.1. Profile Closures
13.2. Distinguishability
Chapter 14. Betweenness
14.1. Wald Betweenness
14.2. Transform Betweenness
14.3. Menger Betweenness
14.4. Probabilistic Betweenness
14.5. Open Problems
Chapter 15. Supplements
15.1. Probabilistic Normed Spaces
15.2. Probabilistic Inner Product Spaces
15.3. Probabilistic Topologies
15.4. Probabilistic Information Spaces
15.5. Generalized Metric Spaces
References
Index
Errata
Notes
Supplementary References
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Preface to the Dover Edition
The republication of our book has given us the opportunity to provide,
not only the customary list of errata (fortunately, nearly all minor), but also
an extensive set of notes and a supplementary bibliography which together
follow the unaltered original text and serve to update it.
The scope of the notes (particularly, those referring to Chapters 5, 6, 7,
11, and 15) and the length of the supplementary bibliography (which is by
no means complete) testify to the remarkable growth of interest in and
applications of several parts of the subject. Many persons have contributed
to this growth: Besides the persons named in the original preface and whose
work is still continuing, we thank the following ones, with whom we have
personally interacted: J. Aczél, F. Balibrea, G. Dall’Aglio, W. F. Darsow,
E. Diday, G.-L. Forti, C. Genest, Tie-Xin Guo, U. Höhle, M. F. Janowitz, E.
P. Klement, G. M. Krause, B. Lafuerza-Guillén, R. B. Nelsen, L. Paganoni,
J. J. Quesada-Molina, T. Riedel, J. A. Rodriguez-Lallena, C. Sempi, J.
Smital, M. D. Taylor, and E. Trillas.
The original edition of our book has been long out of print, yet has been
the subject of increasing demand. We therefore owe special thanks to Dover
Publications, and to its editor John Grafton in particular, for making it again
available in an expanded edition. And for her professional, insightful, and
cheerful assistance in preparing this expanded edition, we thank Kathy
Richards.
Sadly, since 1983, we have lost several friends and colleagues who had
participated in this enterprise. So we pay tribute here to the memory of our
mentor Karl Menger, and also to the memory of Pietro Benvenuti, Alan T.
Bharucha-Reid, Violet Fiatte, Bruno Forte, Joseph Kampé de Fériet, Marek
Kuczma, and György Targonski.
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Preface
The story of this book begins in 1951 when the first author was reading the
book Albert Einstein, Philosopher-Scientist. There, in a paper by Karl
Menger entitled “Modem Geometry and the Theory of Relativity,” he came
across the following passage:
Poincaré, in several of his famous essays on the philosophy of science,
characterized the difference between mathematics and physics as follows: In
mathematics, if the quantity A is equal to the quantity B, and B is equal to C,
then A is equal to C; that is, in modern terminology: mathematical equality is
a transitive relation. But in the observable physical continuum “equal” means
indistinguishable; and in this continuum, if A is equal to B, and B is equal to
C, it by no means follows that A is equal to C. In the terminology of the
psychologists Weber and Fechner, A may lie within the threshold of B, and B
within the threshold of C, even though A does not lie within the threshold of
C. “The raw result of experience,” says Poincaré, “may be expressed by the
relation
which may be regarded as the formula for the physical continuum.” That is
to say, physical equality is not a transitive relation.
Is this reasoning cogent? It is indeed easy to devise experiments which prove
that the question whether two physical quantities are distinguishable cannot
always be answered by a simple Yes or No. The same observer may regard
the same two objects sometimes as identical and sometimes as
distinguishable. A blindfolded man may consider the simultaneous irritation
of the same two spots on his skin sometimes as one, and sometimes as two
tactile sensations. Of two constant lights, he may regard the first sometimes
as weaker than, sometimes as equal to, and sometimes as stronger than the
second. All that can be done in this situation is to count the percentage
number of instances in which he makes any one of these two or three
observations. In the observation of physical continua, situations like the one
just described seem to be the rule rather than the exception.
Instead of distinguishing between a transitive mathematical and an
intransitive physical relation of equality, it thus seems much more hopeful to
retain the transitive relation in mathematics and to introduce for the
distinction of physical and physiological quantities a probability, that is, a
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number lying between 0 and 1.
Elaboration of this idea leads to the concept of a space in which a
distribution function rather than a definite number is associated with every
pair of elements. The number associated with two points of a metric space is
called the distance between the two points. The distribution function
associated with two elements of a statistical metric space might be said to
give, for every x, the probability that the distance between the two points in
question does not exceed x. Such a statistical generalization of metric spaces
appears to be well adapted for the investigation of physical quantities and
physiological thresholds. The idealization of the local behavior of rods and
boards, implied by this statistical approach, differs radically from that of
Euclid. In spite of this fact, or perhaps just because of it, the statistical
approach may provide a useful means for geometrizing the physics of the
microcosm.
Reading this passage crystallized this author’s own thoughts on this topic
and, in short order, led to his coming to the Illinois Institute of Technology to
pursue his graduate studies under Menger’s direction. The second author
joined the faculty of I.I.T. in 1956, and soon thereafter our collaboration on
probabilistic metric spaces began. In the intervening years the subject has
grown into a coherent whole which is the theme of this book.
In our development we follow Menger and work directly with
probability distribution functions, rather than with random variables. This
means that we take seriously the fact that the outcome of any series of
measurements of the values of a nondeterministic quantity is a distribution
function; and that a probability space, although an exceedingly useful
mathematical construct, is in fact and in principle unobservable. This point
of view gives our subject a distinct nonclassical flavor. It has also forced us
to develop a considerable body of new mathematical machinery. Our book
thus divides naturally into two major parts. After the introductory Chapter 1,
in which we give an overview and discuss the historical development of the
theory, we devote Chapters 2–7 to the development of this machinery. Our
presentation here is complete in the sense that all the needed concepts are
defined, all the needed results are stated, and virtually all the proofs not
available in the literature are given.
The study of probabilistic metric spaces proper begins with Chapter 8,
where we give the basic definitions and establish some simple properties.
Chapters 9, 10, and 11 are devoted to special classes of probabilistic metric
spaces; Chapters 12 and 13 to topologies and generalized topologies;
Chapter 14 to betweenness; and the final Chapter 15 to several related
structures, such as probabilistic normed and inner-product spaces.
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