Strong integral of multifunctions relative to a fuzzy measure

In this paper we define and study an integral for measurable multifunctions taking values in the family of all nonvoid subsets of [ 0 , + ∞ ) with respect to a fuzzy measure. Unlike various ways of defining such an integral (e.g. Aumann way using different selections, Dunford way, Riemann way), we use the Sugeno method directly applied to multifunctions. The originality of our definition is that the value of the integral is a real scalar and not usually a set. Some classical integral properties, a convergence theorem and relationships with other integrals are presented.