Fuzzy measure theory

In mathematics, fuzzy measure theory considers generalized measures in which the additive property is replaced by the weaker property of monotonicity. The central concept of fuzzy measure theory is the fuzzy measure (also capacity, see ), which was introduced by Choquet in 1953 and independently defined by Sugeno in 1974 in the context of fuzzy integrals. There exists a number of different classes of fuzzy measures including plausibility/belief measures; possibility/necessity measures; and probability measures, which are a subset of classical measures. Let be a universe of discourse, be a class of subsets of , and . A function where is called a fuzzy measure. A fuzzy measure is called normalized or regular if . A fuzzy measure is: additive if for any such that , we have ; supermodular if for any , we have ; submodular if for any , we have ; superadditive if for any such that , we have ; subadditive if for any such that , we have ; symmetric if for any , we have implies ; Boolean if for any , we have or . Understanding the properties of fuzzy measures is useful in application. When a fuzzy measure is used to define a function such as the Sugeno integral or Choquet integral, these properties will be crucial in understanding the function's behavior. For instance, the Choquet integral with respect to an additive fuzzy measure reduces to the Lebesgue integral. In discrete cases, a symmetric fuzzy measure will result in the ordered weighted averaging (OWA) operator. Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define a Choquet integral. Let g be a fuzzy measure. The Möbius representation of g is given by the set function M, where for every , The equivalent axioms in Möbius representation are: for all and all A fuzzy measure in Möbius representation M is called normalized if Möbius representation can be used to give an indication of which subsets of X interact with one another. For instance, an additive fuzzy measure has Möbius values all equal to zero except for singletons.