The Sugeno integral. A point of view

Abstract

For any positive decreasing function g on the positive real axis, we define a generalized fixed point x0 of g and call x0 “the midpoint of g”. It can be proved that the Sugeno integral of any positive measurable function f with respect to any motonone measure μ is the midpoint of the canonical decumulative function of f and μ. As an application, we obtain an immediate proof of a well-known transformation theorem concerning the Sugeno integral. Sufficient conditions insuring the validity of a popular reduction procedure (using a smaller set than the whole positive real axis) for the computation of the Sugeno integral are given. The preceeding two topics are applied to positive simple functions. At the end, using the theory of midpoints, we show that the Hausdorff dimension and the Hirsch index actually are Sugeno integrals.