On the closure of families of fuzzy measures under eventwise aggregations
The paper discusses several special families of fuzzy measures, especially their closedness under distortion and under aggregation. A special attention is paid to the coherent lower probabilities, \(k\)-monotone fuzzy measures and belief measures (compare also [\textit{L. Valášková} and \textit{P. Struk}, Kybernetika 41, No.~2, 205--212 (2005)]. For example, an aggregation \(A(m_1,\dots,m_n)\) of belief measures \(m_1,\dots,m_n\) is a belief measure independent of the underlying measurable space and incoming belief measures if and only if \(A\) is totally monotone, i.e., if all partial derivatives of any order exist and are non-negative. Observe that by duality, the corresponding results for the dual fuzzy measures can be obtained (e.g., for the coherent upper probabilities or for plausibility measures). Multilinear extensions of fuzzy measures to aggregation operators are also discussed.
scientific article; zbMATH DE number 2175513
scientific article