A Universal Integral as Common Frame for Choquet and Sugeno Integral

The Choquet and the Sugeno integral provide a useful tool in many problems in engineering and social choice where the aggregation of data is required. However, their applicability is restricted because of the special operations used in the construction of these integrals. Therefore, we provide a concept of integrals generalizing both the Choquet and the Sugeno case. For functions with values in the nonnegative real numbers, universal integrals are introduced and investigated, which can be defined on arbitrary measurable spaces and for arbitrary monotone measures. For a fixed pseudo-multiplication on the nonnegative real numbers, the smallest and the greatest universal integrals are given. Finally, another construction method for obtaining universal integrals is introduced, and the restriction to the unit interval, i.e., to fuzzy integrals, is considered.