## 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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addition Archimedean associative binary operation C-space closed closure complete composition consequence consider contains continuous convergence convolution copula Corollary corresponding defined Definition denote determined direction distance distinct distribution functions E-space element equal equivalent established example exist extended fact Figure fixed follows function f Furthermore given Hence holds identity immediate implies independent inequality integer interval introduced Lemma Let S,F margins measure Menger space metric space n-copula natural neighborhood system nondecreasing Note obtain pair particular PM space points positive probabilistic probability probability space Problem PROOF properties pseudometrically Ran f random variables respect satisfies Schweizer sense sequence shows Similarly simple space strict strong topology structure studied subset Suppose t-norm Theorem theory topology transform triangle function unique Wald whence yields