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
Introduction and Historical Survey | 1 |
Distribution Functions | 43 |
Associativity | 54 |
Copyright | |
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a-simple A₁ Archimedean t-norm associative binary operation C₁ canonical E-space closure commutative contraction map convergence convolution copula Corollary d₁ Definition denote distance distinct points distribution functions distribution-generated space equilateral equivalent F and G F₁ following conditions Fpq(x function defined function f G in A+ g₁ given Hence holds idempotent inner product space left continuous Lemma let F Let S,F Let S,F,T Lévy metric Menger space n-box n-copula neighborhood system nondecreasing nonempty nonnegative Note null element ordinal sum pair PM space probabilistic metric spaces probability space Problem profile function PROOF pseudometric space pseudometrically generated space Ran f random variables S₁ satisfies Schweizer Section semigroup sequence Šerstnev simple space strictly increasing strong topology subset Suppose t-norm T₁ transform triangle function triangle inequality v₁ Wald space whence X₁ yields ερ