Fuzzy Measures and Integrals: Recent Developments

This paper gives a survey of the research done on fuzzy measures and integrals since Sugeno proposed in 1974 the concept of fuzzy measure, with an emphasis on recent results. This field of research lies at the intersection of several independent domains, which makes it very active and attractive, namely, measure theory, theory of aggregation functions, cooperative game theory, combinatorial optimization, pseudo-Boolean functions and more generally theoretical computer sciences. As an illustration of this fact, the word “fuzzy measure” which was coined by Sugeno, has many different names according to the field where it is used: nonadditive measure, capacity, monotone game, pseudo-Boolean function, rank function of a polymatroid, etc. Evidently, this short paper cannot make a complete account of all the research undertaken in this area, a whole book would hardly suffice. Indeed, the author is preparing a monograph on this topic, with the title: “Set functions, games and capacities in decision making”, to be published by Springer around the end of 2015. This paper gives a kind of quick and necessarily simplified summary of selected topics. We recommend the interested reader to consult the main (available) monographs dealing with fuzzy measures and integrals: Pap [46], Denneberg [11], Wang and Klir [63], the Handbook of measure theory edited by Pap [47], as well as the edited book [24], and the survey paper [22]. The latter focusses on application in multicriteria decision making, an important field of applications which is not covered by this paper, restricting to theory. To avoid intricacies, in the whole paper the universal set X is finite, with |X| = n. We often use ∨,∧, which collapse to maximum and minimum on finite sets.