ON A NON-LINEAR INTEGRAL OF MULTIFUNCTIONS WITH RESPECT TO A FUZZY MEASURE ANCA CROITORU ”Al.I. Cuza” University, Faculty of Mathematics, Bd. Carol I, No. 11, Ias ¸i, 700506, ROMANIA croitoru@uaic.ro Abstract: We present some properties of fuzzy mean convergence regarding a non-linear integral of measurable multifunctions with respect to a fuzzy measure. Key–Words: non-linear integral, fuzzy measure, measurable multifunction. 1 Introduction Non-additive measure theory has been investigated by many authors (e.g., Aumann [1], Choquet [2], Croitoru et al. [5], Debreu [6], Drewnowski [7], Gavrilut ¸ [10,11], Gavrilut ¸ and Croitoru [12], Guo and Zhang [13], Jang and Kwon [16], Martellotti and Sambucini [17], Pap [18], Precupanu [19], Pre- cupanu and Croitoru [20], Precupanu, Gavrilut ¸ and Croitoru [21], Ralescu and Adams [22], Ralescu and Sugeno [23], Roman-Flores, Flores-Franulic and Chalco-Cano [24], Stamate [27], Sugeno [28], Suzuki [29], Wang and Klir [30], Zhang and Guo [34]), due to its applications in statistics, economics or theory of control. The non-linear integrals of Choquet [2] or Sugeno [28] are used in statistics ([15]) in relation to robust- ness problems in testing of hypothesis, in Bayesian robustness ([31], [32]), in mathematical economics in connection to an extension of utility theory ([25], [26], [33]), in aggregation problems or in human evaluation processes ([28]). In [4] we have defined a non-linear integral for measurable multifunctions with respect to a fuzzy measure. Such an integral may be used in synthetic evaluation of the quality of a given object, when the score function may be set-valued i.e., for each quality factor there exists a multiple score or a set of estima- tions. In this paper we present some properties of fuzzy mean convergence regarding the non-linear integral introduced in [4]. In section 2, the preliminary definitions have been introduced and the non-linear integral of [4] has been reminded. In section 3 we give some results concerning the fuzzy mean convergence and its relationships with other types of convergences such as convergence in fuzzy measure and almost everywhere convergence. 2 Preliminary definitions For a non-empty set X , P 0 (X ) is the family of non- empty subsets of X . Let (X, d) be a metric space. Then P f (X ) is the family of non-empty closed subsets of X , P bf (X ) is the family of non-empty closed bounded subsets of X and P k (X ) is the family of non-empty compact subsets of X . For every M,N ∈P 0 (X ), we de- note h(M,N ) = max {e(M,N ),e(N,M )}, where e(M,N )= sup x∈M d(x, N ) is the excess of M over N and d(x, N ) = inf y∈N d(x, y) is the distance from x to N. It is known that h becomes an extended metric on P f (X ) (i.e. it is a metric which can also take the value +∞) and h becomes a metric (called the Hausdorff metric) on P bf (X ) (Hu and Papageorgiou [14]). We denote R + = [0, +∞), N * = N\{0}. Now, let X be a real linear space. On P 0 (X ) we consider an order relation denoted by ” ≤ ” and we shall write (P 0 (X ), ≤). For convenience, the notation F ≥ E will be used instead of E ≤ F , for E,F ∈ P 0 (X ). Example 2.1. I. The usual set inclusion ”⊆” is an or- der relation on P 0 (X ) and we write (P 0 (X ), ⊆). II. Let (X, ≤) be a real ordered vector space. For every E,F ∈P 0 (X ) and α ∈ R let E + F = {x + y|x ∈ E,y ∈ F }, Recent Researches in Computational Techniques, Non-Linear Systems and Control ISBN: 978-1-61804-011-4 79