Chapter 33 - Monotone Set Functions-Based Integrals

Abstract

The Lebesgue integral is a monotone nonnegative linear function on the space of bounded measurable functions; its construction is strongly linked with the additivity of the underlying measure. However, the additivity of a measure is rather restrictive when modeling several real situations. To overcome this problem, several generalizations of classical probability measures have been proposed, such as pseudoadditive measures, possibility measures, belief functions, k-monotone capacities, and k-order additive measures. All these generalizations are covered by fuzzy measures. These set functions are sometimes called also premeasures. The Choquet integral is defined for nonnegative functions, and it is comonotone additive and homogeneous with respect to the multiplication by nonnegative reals. The Sugeno integral deals with functions whose range is contained with fuzzy sets and with normed fuzzy measures. The properties of the Choquet and the Sugeno integrals are recalled, discussed, and compared in this chapter. The chapter introduces a class of general fuzzy integrals with respect to a general fuzzy measure that has properties similar to those of Choquet or Sugeno integrals. To express the comonotone additivity of the Choquet integral and the comonotone maxitivity of the Sugeno integral, it is convenient to introduce a representation by means of a system of comonotone basic functions. The pseudoadditions and pseudomultiplications are also elaborated in the chapter.