Comonotone aggregation operators (Q2719815)

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 \(\mu_a(A)= L(a\cdot 1_A)\).