A Generalization of the Choquet Integral Defined in Terms of the Möbius Transform

In this work we propose a generalization of the Choquet integral starting from its definition in terms of the Mobius transform. We modify the product on the set of real numbers considered in the Lovasz extension form of the Choquet integral into a function $F$ , and we discuss the properties of this new functional. For a fixed $n$ , a complete description of all $F$ yielding an $n$ -ary aggregation function with a fixed diagonal section, independently of the considered fuzzy measure, is given, and several particular examples are presented. Finally, all functions $F$ yielding an aggregation function, independently of the number $n$ of inputs and of the considered fuzzy measure, are characterized, and related aggregation functions are shown to be just the Choquet integrals over the distorted inputs.