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Comonotone aggregation operators. (English) Zbl 0999.28011
Let $$(\Omega,{\mathcal A})$$ be a measurable space, $$0< F\leq+\infty$$, $${\mathcal F}$$ the family of all $${\mathcal A}$$-measurable $$[0,F]$$-valued functions and $$\oplus$$ a pseudo-addition on $$[0,F]$$. A $$\oplus$$-comonotone aggregation operator is defined as a functional $$L:{\mathcal F}\to [0,F]$$ which is idempotent, monotone, continuous from below and comonotone $$\oplus$$-additive, i.e., $$L(f_1\oplus f_2)= L(f_1)\oplus L(f_2)$$ if $$f_1$$ and $$f_2$$ are comonotone. The main theorem gives a bijection between the space of all $$\oplus$$-comonotone aggregation operators on $${\mathcal F}$$ and a certain class of families $$\{\mu_a:0< a\leq F\}$$ of fuzzy measures $$\mu_a: A\to [0,F]$$. The fuzzy measures $$\mu_a$$ associated to $$L$$ are defined by $$\mu_a(A)= L(a\cdot 1_A)$$.
Reviewer: Hans Weber (Udine)

##### MSC:
 2.8e+11 Fuzzy measure theory