Silvano Holzer

Chebyshev type inequality for Choquet integral and comonotonicity

We supply a Chebyshev type inequality for Choquet integral and link this inequality with comonotonicity.

A Chebyshev type inequality for Sugeno integral and comonotonicity

We supply a characterization of comonotonicity property by a Chebyshev type inequality for Sugeno integral.

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; afterwards they 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.

Some remarks on T-supermodularity of Choquet integral

We prove that total monotonicity of monotone measures is a sufficient (but not necessary) condition for T-supermodularity of Choquet integral. Moreover, we show that total monotonicity does not imply, in general, supermodularity of the integral, when we consider the symmetric Choquet integral or the Sugeno integral. Finally, we also prove that, for the Choquet integral, T-supermodularity implies supermodularity, when T is the product t-norm, and it is equivalent to supermodularity, when T is the Łuckasiewicz t-norm.

Regular and purely irregular bounded charges: a decomposition theorem

We introduce the notions of regular and purely irregular charges with respect to a pair of pavings and study their structural properties. Moreover, we link regularity and σ-additivity, obtaining some generalizations of well-known theorems. Finally, when the pavings satisfy some reasonable weak conditions, we can decompose any bounded charge into regular and purely irregular decomposants; this decomposition becomes the Hewitt-Yosida one, whenever the charges are defined on the Baire σ-field of a countably compact space

A remark on failures of conglomerability of prevision

Given a coherent conditional prevision we obtain, by means of the Yosida-Hewitt Decomposition Theorem, an upper bound of its failures of conglomerability with respect to a random number and a denumerable partition. In this way we get an extension of Theorem 2.3 of [4] to coherent previsions.RiassuntoConsiderata una previsione subordinata coerente si fornisce, tramite il teorema di decomposizione di Yosida-Hewitt, una maggiorazione degli errori di conglomerabilità relativi ad un numero aleatorio e ad una partizione numerabile. Viene così esteso alle previsioni coerenti il teorema 2.3 di [4].