## Probabilistic Metric SpacesThis distinctly nonclassical treatment focuses on developing aspects that differ from the theory of ordinary metric spaces, working directly with probability distribution functions rather than random variables. The two-part treatment begins with an overview that discusses the theory's historical evolution, followed by a development of related mathematical machinery. The presentation defines all needed concepts, states all necessary results, and provides relevant proofs. The second part opens with definitions of probabilistic metric spaces and proceeds to examinations of special classes of probabilistic metric spaces, topologies, and several related structures, such as probabilistic normed and inner-product spaces. Throughout, the authors focus on developing aspects that differ from the theory of ordinary metric spaces, rather than simply transferring known metric space results to a more general setting. |

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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.

Huse Fatkić

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Alsina Archimedean t-norm binary operation canonical E-space closure operation commutative continuous t-norm contraction map convergence convex convolution copula Corollary deﬁned Definition denote distance distinct points distribution functions distribution-generated space edition equivalent Euclidean example F and G ﬁrst following conditions FPq(x function defined function f geometry given Hence holds idempotent inner product space interval Kuratowski closure operation left continuous Lemma Let f Lévy metric linear Math mathematical measure Menger space metric transform Moynihan n-copula neighborhood system nondecreasing nonempty nonnegative Note null element ordinal sum pair PM space probabilistic metric spaces probability space Problem PROOF properties pseudometrically generated space quantum quasi-inverse Ran f random metric space random variables S X S satisfies Schweizer Section semigroups sequence Serstnev Sherwood simple space Sklar statistical strictly increasing strong topology subset Suppose t-conorm t-norm theory tion triangle function triangle inequality Univ Wald space whence yields