CHAPTER 34 – Set Functions over Finite Sets: Transformations and Integrals

This chapter elaborates various transformations of the set functions. Measures in a finite setting are positive-valued set functions with some characteristic properties. It is found that although the finiteness of the universe considerably restricts the interest of the measure concept, it also allows other viewpoints coming from various fields of mathematics—suchas combinatorics, game theory, and complexity—when viewing measures as particular set functions or pseudo-Boolean functions. There is a general formula for the inverse of cardinality operators. The formalism permits to compute easily, simply by combining and inversing operators, all formulas between the different representations of set functions. It is found that any set function can be decomposed on the set of unanimity games or their conjugate. The k-order additive measures or k-additive measures decrease the exponential complexity of fuzzy measures in practical applications. The ordinal interaction transform of a fuzzy measure is also elaborated in the chapter.

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