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1
Introduction and Historical Survey
1.0. Introduction
The 19th century, which marks the beginning of the modern age of science, was an era of great advances in the art of measurement. These advances stimulated a corresponding concern with the accompanying errors. Until the early part of this century, however, it was still believed that through careful design and ample data the error in any measurement could be made arbitrarily small. The advent of quantum mechanics shattered this belief, for here the uncertainties of the measurements are inherent in the measurement process itself and in principle cannot be removed.
Today, in the fourth quarter of the 20th century, the existence of such inherent uncertainties and thresholds is a generally accepted fact. This is true not only in physics but also in areas such as psychometrics [S. Stevens 1959], communication theory [Shannon 1948; Brillouin 1956], and pattern recognition [Duda and Hart 1973; Yakimovsky 1976; Prager 1979]. It is also central to various mathematical disciplines, such as cluster analysis [Janowitz 1978; Shepard 1980] and interval analysis [Moore 1979]. However, in virtually all the mathematical models built to describe these various situations it is assumed that the measurements in question are made with respect to a rigid reference frame. Remarks to the effect that this assumption may be unsatisfactory and that some of the uncertainties should be built into the geometry are scattered here and there in the literature [Poincaré 1905, 1913; Hjelmslev 1923; de Broglie 1935; Black 1937; Weyl 1952; Bom 1955; Oppenheimer 1962], along with suggestions on the proper way of doing this [Penrose and MacCullum 1973; Penrose 1975]. There are also some serious attempts in this direction [Eddington 1953; Rosen 1947, 1962; Blokhintsev 1971, 1973; Frenkel 1977]. This book is a direct outgrowth of one of these attempts, namely, the theory of probabilistic metric spaces as initiated by K. Menger in 1942.
1.1. Beginnings
The first abstract formulation of the notion of distance is due to M. Fréchet [1906]. This notion, which was later given the name “metric space” (“metrischer Raum”) by F. Hausdorff [1914], is based on the introduction of a function d that assigns a nonnegative real number d(p, q) (the distance between p and q) to every pair (p, q) of elements (points) of a nonempty set S. This function is assumed to satisfy the following conditions:
Condition (1.1.4), whose antecedents go back at least to Euclid’s Proposition I.20, is the triangle inequality.
Any function d satisfying (1.1.1)–(1.1.4) is a metric on S. (Occasionally it is convenient to drop (1.1.2), in which case d is a pseudometric on S.) A metric space is a pair (S, d) where S is a set and d is a metric on S.
In 1942 K. Menger, who had played a major role in the development of the theory of metric spaces (see [Menger 1928, 1930, 1932, 1954]), proposed a probabilistic generalization of this theory. Specifically, he proposed replacing the number d(p, q) by a real function Fpq whose value Fpq(x), for any real number x, is interpreted as the probability that the distance between p and q is less than x. Since probabilities can neither be negative nor be greater than 1, we have
for every real x, and clearly we also have
whenever x < y. Hence Fpq is a probability distribution function.
Other conditions on the functions Fpq are also immediate. Thus, since distances cannot be negative, we have
Similarly, (1.1.1), (1.1.2), and (1.1.3) yield the following:
In contradistinction, the probabilistic generalization of the triangle inequality (1.1.4) is quite another matter. Even at the outset Menger and A. Wald proposed different generalizations, and the study of alternative triangle inequalities has been a central theme in the development of the theory of probabilistic metric spaces. The major steps in this development are traced in the next few sections.
1.2. Menger, 1942
In his original paper, Menger [1942] defined a statistical metric space1 as a set S together with an associated family of probability distribution functi...
