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. |
Other editions - View all
Common terms and phrases
a-simple Alsina Archimedean t-norm associative binary operation C-space continuous t-norm contraction map convergence convolution copula Corollary d₁ d₂ Definition denote distance distinct points distribution functions distribution-generated space edition equivalent Euclidean example F and G F₁ Fpq(x functions F geometry given Hence idempotent inner product space integer Kuratowski closure operation left continuous Lemma Let S,F,T Lévy metric linear Math mathematical measure Menger space metric transform n-copula neighborhood system nondecreasing nonempty normed spaces Note null element ordinal sum pair PM space probabilistic metric spaces probability space Problem PROOF pseudometrically generated space Ran f random normed random variables S₁ S₂ satisfies Schweizer Section semigroups sequence Šerstnev Sherwood simple space Sklar statistical strictly increasing strong topology subset t-conorm t-norm T₁ T₂ Theorem theory tion triangle function triangle inequality TT,L Wald space whence yields