Gabriele H. Greco

Fuzzy integrals and fuzzy measures with their values in complete lattices

Abstract Real-valued fuzzy measures and their associated fuzzy integrals were proposed by Sugeno. In this paper we consider fuzzy measures valued in complete lattices and their associated upper and lower fuzzy integrals. Moreover we define a class of functionals, called fuzzy integrals. In the setting of the complete lattices which are both Brouwerian and dually Brouwerian, the upper and lower fuzzy integrals are fuzzy integrals; while in that of the completely distributive complete lattices every fuzzy integral is both lower and upper fuzzy integral with respect to a fuzzy measure.

Decomposizioni di semifiltri e Γ-limiti sequenziali in reticoli completamente distributivi

SummarySequential forms of De Giorgis Γ-limits are obtained via a decomposition of their carriers (carrier=sostegno) in the setting of completely distributive complete lattices. Moreover, the notion of carrier enables us to calculate easily the K-limits of epi- or hipo-graphs of arbitrary Γ-limits.

Minimax theorems and saddling transformations

Abstract A transformation mapping games onto games that have saddle values (for short, saddling transformation) was first considered by von Neumann. By introducing mixed strategies, he showed that the bilinear extension of a finite game has a saddle value. This paper is concerned with saddling transformations of topological type in the setting of the quasi concave-convex payoffs and with their compositions. It yields new minimax theorems which are extensions of the Sion theorem.

On L-fuzzy integrals of measurable functions

Abstract It is shown that the L-fuzzy integral associated with an L-fuzzy measure α (where L is a completely distributive complete lattice) when calculated for functions measurable with respect to a family of sets H depends only on the restriction of α to H . This result is used to obtain some semicontinuity properties of L-fuzzy integrals.

On functionals representable by fuzzy measures

Necessary and sufficient conditions for a functional T to be representable by fuzzy-measure are given. The calculus of the minimal αT and the maximal βT fuzzy-measures which represent the functional T and the fuzzy-integrals f ( · ) dαT and f ( · ) dβT are made explicit in terms of T. Moreover, it is shown that, in case T is defined on a fuzzy-domain of integration B, every f ϵ B is RT-measurable. Finally some functionals related to fuzzy additive measures are presented.

Un teorema di Radon-Nikodym per funzioni d’insieme subadditive

RiassuntoIn questo articolo si danno delle condizioni necessarie e sufficienti affinchè per una fissata coppia di funzioni d’insieme ν, μ crescenti esista una funzionef tale che ν=∫fdμ. Si ottiene cosi una proposizione comprendente il teorema di R-N. classico e dei teoremi di R-N., presentati da altri autori, riguardanti le funzioni d’insieme finitamente additive e le funzioni d’insieme subadditive e continue per successioni crescenti.RésuméSoient ν, μ:A→[0,+∞) deux fonctions d’ensemble croissantes sur une σ-algèbre d’ensemblesA⊂T(X), telles que pour chaqueA∈A avec ν(A)=μ(A)=0 on a l’égalité μ(A)=μ(A∪S) ∀S∈A (c’est le cas des fonctions sousadditives!). Dans cet article on démontre qu’il existe une fonctionfA-measurable telle que ν=∫fdμ si et seulement si pour chaquer∈(0, + ∞) il y a un ensembleAr∈A qui vérifie les trois conditions suivantes: (1)

Reconstruction of polytopes by convex pastings

v\left( A \right) - \mu \left( B \right) > r\left( {v\left( A \right) - \mu \left( B \right)} \right)\forall A

Topological versions of passy– prisman’S minimax theorem

,B∈A avecB⊂A; (2)