A characterization of neo-additive measures

Neo-additive and generalized neo-additive capacities were introduced in order to capture both optimistic and pessimistic attitudes towards uncertainty without abandoning the subjective probabilistic approach. In this way, one can obtain, as particular cases, some well-known decision criteria (via Choquet expectation) adopted in Decision Theory and Mathematical Statistics. In order to introduce these capacities, Chateauneuf, Eichberger, Grant and Eichberger, Grant, Lefort consider three types of events: universal, null and essential events; afterwardsthey introduce capacities which are null on null events (null property), assume value one on universal events (normalization property) and are translations of finitely additive probabilities on the family of essential events. Finally, they supply a theoretic measure characterization of these type of capacities. In this paper, we introduce neo-additive measures as monotone measures which are translations of finitely additive ones on the family of essential events, without assumption of normalization property and null property. Moreover, we supply a simple and natural theoretic characterization of these measures obtaining, as particular cases, the corresponding results of the previous authors. In this way, our results give a robust foundation of neo-additive and generalized neo-additive capacities in abstract measure setting.