## 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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a-simple allp Alsina Archimedean t-norm associative 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 function f G in A+ geometry given Hence holds idempotent implies inner product space interval Kuratowski closure operation left continuous Lemma 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 qu(x 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 Wald space whence yields