zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Probabilistic metric spaces. (English) Zbl 0546.60010
North Holland Series in Probability and Applied Mathematics. New York-Amsterdam-Oxford: North-Holland. XVI, 275 p. $ 39.00; Dfl. 120.00 (1983).

K. Menger [Statistical metrics. Proc. Natl. Acad. Sci. USA 28, 535- 537 (1942)] proposed a probabilistic generalization of the theory of metric spaces by introducing the concept of probabilistic (statistical) metric space. This paper by Menger constituted the starting point for a field of research known as the theory of probabilistic metric spaces. This monograph presents an organized body of advanced material on this theory, incorporating much of the authors’ own research.

It begins with the introductory chapter 1 devoted to historical aspects of this theory. The remaining chapters are divided into two major parts. chapters 2 through 7 develop the mathematical tools which are needed for the study of probabilistic metric spaces. This study properly begins with chapter 8 and goes through to chapter 15. Chapter 8 contains the basic definitions and simple properties. Chapters 9, 10, and 11 are devoted to special classes of probabilistic spaces: random metric spaces, distribution-generated spaces, and transformation-generated spaces. Chapters 12 and 13 deal with topologies and generalized topologies. Chapter 14 is devoted to betweenness. The final chapter is concerned with related structures such as probabilistic normed, inner-product, and information spaces. An extensive literature accompanies the text.

Clearly written, this unified and self-contained monograph on probabilistic metric spaces will be particularly useful to researchers who are interested in this field. It is also suitable as a text for a graduate course on selected topics in applied probability.

Reviewer: R.Theodorescu

60B99Probability theory on general structures
54E35Metric spaces, metrizability
60-02Research monographs (probability theory)
60E05General theory of probability distributions
60B05Probability measures on topological spaces
54E99Topological spaces with richer structures