On the integration of fuzzy level sets

Charles Castaing (1) , Christiane Godet-Thobie (2) , Thi Duyen Hoang (3) , Paul Raynaud de Fitte (4)

1 UM2 - Université Montpellier 2 - Sciences et Techniques
2 LM - Laboratoire de mathématiques de Brest
3 Quang Binh University
4 LMRS - Laboratoire de Mathématiques Raphaël Salem

We study the integration of fuzzy level sets associated with a fuzzy random variable when the underlying space is a separable Banach space or a weak star dual of a separable Banach space. In particular, the expectation and the conditional expectation of fuzzy level sets in this setting are presented. We prove the SLLN for pairwise independent identically distributed fuzzy convex compact valued level sets through the SLLN for pairwise independent identically distributed convex compact valued random set in separable Banach space. Some convergence results for a class of integrand martingale are also presented.

Mots clés: conditional expectation; fuzzy convex; fuzzy martingale; integrand martingale; level set; upper semicontinuous

Domaines: Mathématiques [math]

Charles Castaing, Christiane Godet-Thobie, Thi Duyen Hoang, Paul Raynaud de Fitte. On the integration of fuzzy level sets. Advances in Mathematical Economics, Springer, 2015, ⟨10.1007/978-4-431-55489-9_1⟩. ⟨hal-01784853⟩