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. |
Contents
Distribution Functions | 4 |
Preliminaries | 18 |
Metric and Topological Structures | 30 |
Copyright | |
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a-simple Archimedean t-norm associative binary operation canonical E-space contraction map convergence convolution copula Corollary d₁ d₂ Definition denote distance distinct points distribution-generated space equivalent F and G F₁ following conditions Fpq(x function defined function f given Hence idempotent identity integer Kuratowski closure operation left continuous Lemma let F Let S,F,T Lévy metric Math Menger space metric transform Moynihan n-copula neighborhood system nondecreasing nonempty nonnegative normed spaces Note null element ordinal sum pair PM space probabilistic metric spaces probability space Problem PROOF properties pseudometric space pseudometrically generated space quasi-inverse Ran f random metric space random variables S₁ satisfies Schweizer Section semigroups sequence Šerstnev Sherwood simple space Sklar strictly increasing strong topology subset Suppose t-norm T₁ T₂ theory triangle function triangle inequality TT.L Wald space whence X₁ yields