Norms and Discrete Choquet Integrals Induced by Submodular Fuzzy Measures: A Discussion.

The Choquet integral is a powerful tool in multi-criteria decision making and decision under uncertainty. This paper studies the use of its discrete form for the definition of norms, in the general case beyond the often considered case of Ordered Weighted Averages. It proposes a discussion of the characterisation based on Metric Inducing Fuzzy Measures (MIFM) introduced by Bolton et al., 2008, questioning its results. It then describes a characterisation for the discrete case that relates to the notion of properties holding almost everywhere derived from the null sets associated to a fuzzy measure. It discusses in particular the case of Choquet integrals induced by possibility measures.