## 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. |

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a-simple addition binary operation closure operation commutative contraction map convergence convex convolution copula Corollary Definition denote determined distance distinct points distribution functions distribution-generated space E-space equilateral equivalent Euclidean example exist F and G following conditions function defined functions F Furthermore G in A+ given Hence holds hypotheses of Theorem idempotent identity implies isometric Kuratowski closure operation left continuous Lemma measure Menger space metric transform Moynihan n-copula neighborhood system nondecreasing nonempty nonnegative norm space normal C-spaces Note null element operation on A+ ordinal sum pair PM space probabilistic metric spaces probability space Problem profile function proof properties pseudometric space pseudometrically generated space quasi-inverse Ran g random variables S X S Schweizer Section semigroup Serstnev Sherwood simple space space briefly strictly increasing strong topology subset Suppose t-norm Theorem transformation-generated space triangle function triangle inequality Wald space whence yields