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**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; 0063.03886)] 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.

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: Radu Theodorescu (Quebec)

### MSC:

60B99 | Probability theory on algebraic and topological structures |

54E70 | Probabilistic metric spaces |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60E05 | Probability distributions: general theory |

60B05 | Probability measures on topological spaces |

54E35 | Metric spaces, metrizability |