On the axiomatic treatment of the ϕ-mean

Summary We extend to masses on a real interval the notion of ϕ-mean, usually considered in the context of σ-additive probabilities or probability distribution functions, and consider some axiomatic treatments of it at different levels of masses (simple masses, compact support masses, tight masses, arbitrary masses). Moreover, as an important special case, we get axiomatic systems for general means, as well. We also prove that the usual axiomatic system “Consistency with Certainty+Associativity+Monotonicity” characterizes the ϕ-mean of masses with arbitrary compact support and that, already at tight masses level, this system is not adequate. We note that the analytical tool used to define the ϕ-mean is the Choquet integral. Work performed under the auspices of the National Group: “Inferenza Statistica: basi probabilistiche e sviluppi metodologici” (MURST 40%).