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Zbl 0546.60010
Schweizer, B.; Sklar, A.
Probabilistic metric spaces. (English)
[B] North Holland Series in Probability and Applied Mathematics. New York-Amsterdam-Oxford: North-Holland. XVI, 275 p. {\$} 39.00; Dfl. 120.00 (1983).

{\it 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. \par 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. \par 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.
[R.Theodorescu]

MSC 2000:: *60B99 Probability theory on general structures
54E35 Metric spaces, metrizability
60-02 Research monographs (probability theory)
60E05 General theory of probability distributions
60B05 Probability measures on topological spaces
54E99 Topological spaces with richer structures

Keywords: probabilistic (statistical) metric space; random metric spaces

Cited in: Zbl 1231.60008 Zbl 1240.40032 Zbl 1180.54028 Zbl 1162.54013 Zbl 1148.26016 Zbl 1118.54024 Zbl 1080.54022 Zbl 1064.62060 Zbl 1077.39021 Zbl 1052.54034 Zbl 1037.62047 Zbl 1020.60018 Zbl 0972.03026 Zbl 0956.54018 Zbl 0939.15011 Zbl 0939.54015 Zbl 0939.54016 Zbl 0914.04006 Zbl 0943.54002 Zbl 0812.60022 Zbl 0893.54026 Zbl 0866.54030 Zbl 0694.60002 Zbl 0675.54027 Zbl 0616.60011 Zbl 0587.39006 Zbl 0587.39005 Zbl 0549.26013 Zbl 0586.47063

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