Measures on Tribes of Fuzzy Sets and the Role of Frank t-norms

As a natural generalization of a probability space, Butnariu and Klement introduced T-tribes of fuzzy sets with probability T-measures. Their complete characterization was found for the case when T is a Frank t-norm. Here we characterize T-measures with respect to non-Frank strict t-norms. We show that Frank t-norms play a special role in these questions. 1 Generalization of probability spaces Before we describe the approach of 5, 8], we brieey summarize the classical deenition of a probability space. We sometimes use diierent (but equivalent) sets of axioms which are closer to the intended generalization. A. We start from a universal set, X. B. We take a-algebra B of subsets of X as the event structure. Axiomatically, B 2 X such that T n2N A n 2 B. C. We deene a probability measure as a function m: B ! 0; 1] satisfying the axioms The author gratefully acknowledges the support of the project no. VS96049 of the Czech Ministry of Education , the grant no. 201/97/0437 of the Grant Agency of the Czech Republic, the project Aktion Osterreich { Tschechis-che Republik 16p12 and COST Action 15. where the symbol % denotes monotone increasing convergence. D. The triple (X; B; m) is called a probability space. Following 5, 8], a fuzzy generalization of probability space is deened as follows: A. We start from a universal set, X. B1. We take a-algebra B of subsets of X. B2. We deene the collection T of all B-measurable fuzzy subsets of X, i. e., T consists of all functions A: X ! 0; 1] such that A ?1 ((r; 1]) 2 B for all r 2 0; 1]. (We identify fuzzy sets with their membership functions.) The collection T is called the B-generated tribe. C1. We x a t-norm, i. e., a binary operation T: 0; 1] 2 ! 0; 1] which is commutative, associative, nondecreasing, and satisses the boundary condition T(a; 1) = a for all a 2 0; 1] (see 18]). We take the standard fuzzy negation 0 : 0; 1] ! 0; 1] deened by a 0 := 1 ? a, and the t-conorm S: 0; 1] 2 ! 0; 1] dual to T, i. e., S(a; b) := T(a 0 ; b 0) 0. C2. We extend the operations T, 0 , S to operations T, c , S on T (i. e., on fuzzy sets) pointwise: T(A; B)(x) = …