NULL-ADDITIVE FUZZY SET MULTIFUNCTIONS

NULL-ADDITIVE FUZZY SET MULTIFUNCTIONS ALINA GAVRILUŢ Al.I. Cuza University, Faculty of Mathematics, Bd. Carol I, No. 11, Iaşi, 700506, ROMANIA gavrilut@uaic.ro ANCA CROITORU Al.I. Cuza University, Faculty of Mathematics Bd. Carol I, No. 11, Iaşi, 700506, ROMANIA croitoru@uaic.ro NIKOS E. MASTORAKIS Military Institutes of University Education (ASEI), Hellenic Naval Academy Terma Hatzikyriakou, 18593, GREECE mastor@wseas.org Abstract: In this paper we present some properties concerning non-atomicity of regular null-additive fuzzy set multifunctions. Key words: fuzzy, null-additive, regular, atom, non-atomic. 1 Introduction Since Zadeh [28] introduced fuzzy sets in 1965, the theory of fuzzyness began to develop thanks to its applications in probabilities (e.g. Dempster [3], Shafer [25]), computer and systems sciences, artifical intellingence (e.g. Mastorakis [17]), physics, biology, medicine (e.g. Pham, Brandl, Nguyen N.D. and Nguyen T.V. [23]). In the last decades, many authors (e.g. Denneberg [4], Dinculeanu [5], Drewnowski [6], Narukawa [19], Olejček [20], Pap [21,22], Precupanu [24], Suzuki [26], Wu and Bo [27]) investigated the non-additive measure theory due to its applications in mathematical economics, statistics or theory of games (e.g. Aumann and Shapley [1]). In some of our previous articles ([2],[7-15],[18]) we introduced and studied different concepts (such as atom, pseudo-atom, regularity, Darboux property) in set-valued case. In this paper we present some properties concerning non-atomicity of regular null-additive fuzzy set multifunctions defined on the Baire (or Borel) δ-ring of a Hausdorff locally compact space and taking values in P f (X), the family of nonvoid closed subsets of a real normed space X. 2 Preliminary definitions and remarks We now introduce the notations and several definitions used throughout the paper. Let T be a Hausdorff locally compact space, C a ring of subsets of T, B 0 the Baire δ-ring generated by the G δ -compact subsets of T (that is, compact sets which are countable intersections of open sets) and B the Borel δ-ring generated by the compact subsets of T. It is well known that B 0 B (see [5]). X will be a real normed space, P 0 (X) the family of all nonvoid subsets of X, P f (X) the family of all nonvoid closed subsets of X, P bf (X) the family of all nonvoid closed bounded subsets of X and h the Hausdorff pseudometric on P 0 (X): h(m, N) = max{e(m, N), e(n, M)}, where e(m, N) = sup d(x, N), for every M, N x M P 0 (X) and d(x, N) is the distance from x to N with respect to the distance induced by the norm of X. Obviously, e(m, N) = 0 for every M, N P f (X) with M N. It is known that h becomes an extended metric on P f (X) (i.e. a metric that can also take the value + ) and h becomes a metric on P bf (X) ([16]). We define M = h(m, {0}), for every M P 0 (X), where 0 is the origin of X. On P 0 (X) we consider the Minkowski addition

+, defined by: M + N = M + N, for every M, N P 0 (X), where M + N is the closure of M + N with respect to the topology induced by the norm of X. We denote N = N\{0}. Definition 2.1. [2] Let µ : C P 0 (X) be a set multifunction. µ is said to be: I) monotone if µ(a) µ(b), for every A, B C, with A B. II) fuzzy if µ is monotone and µ( ) = {0}. III) null-additive if µ(a B) = µ(a), for every A, B C, with µ(b) = {0}. IV) null-null-additive if µ(a B) = {0}, for every A, B C, with µ(a) = µ(b) = {0}. Remark 2.2. I) Any null-additive set multifunction is null-null-additive. The converse is not valid. Indeed, let T = {a, b}, C = P(T ) and µ : C P f (R), defined by µ(t ) = [0, 2], µ({a}) = µ( ) = {0} and µ({b}) = [0, 1]. Then µ is null-null-additive, but it is not null-additive. II) Let T = [0, + ), C = P(T ) and µ : C P 0 (R) defined by µ( ) = {0}, µ(a) = A if carda = 1, µ(a) = [0, δ(a)] if A is bounded with carda 2 and µ(a) = [0, + ] if A is not bounded (here, carda is the cardinal of A and δ(a) = sup{ t s ; t, s A} is the diameter of A). Then µ is null-additive and nonfuzzy. Definition 2.3. [18] Let µ : C P 0 (X) be a set multifunction. I) A set A C is said to be an atom of µ if µ(a) {0} and for every B C, with B A, we have µ(b) = {0} or µ(a\b) = {0}. II) µ is said to be non-atomic if it has no atoms. III) µ is said to be semi-convex if for every A C, there is B C, with B A so that µ(b) = 1 2 µ(a). Example 2.4. I) Let T = {2n n N}, C = P(T ) and the fuzzy set multifunction µ : C P 0 (R) defined by: { {0}, if A = µ(a) = 1 2A {0}, if A were we denoted 1 2 A = { 1 2x x A}. If A C, A {0} and carda = 1 or A = {0, 2n}, n N, then A is an atom of µ. If A C, A has carda 2 and there exist a, b A such that a b and ab 0, then A is not an atom of µ. II) Let T = [0, 3], C the Borel σ-algebra of T, λ : C [0, + ) the Lebesgue measure and the fuzzy set multifunction µ : C P f (R) defined, for every A C, by: { [ λ(a), λ(a)], if λ(a) 1 µ(a) = [ λ(a), 1], if λ(a) > 1. Then µ is non-atomic. III) Let T = [0, + ), C the Borel σ-algebra of T, λ : C [0, + ) the Lebesgue measure and the fuzzy set multifunction µ : C P f (R) defined by µ(a) = [0, λ(a)], for every A C. Then µ is semiconvex. Definition 2.5. [10] Let A C be an arbitrary set and µ : C P f (X) a set multifunction, with µ( ) = {0}. I) A is said to be: (i) R - regular with respect to µ if for every ε > 0, there exist a compact set K A, K C and an open set D A, D C such that h(µ(a), µ(b)) < ε, for every B C, K B D. (ii) R l - regular with respect to µ if for every ε > 0, there exists a compact set K A, K C such that h(µ(a), µ(b)) < ε, for every B C, K B A. (iii) R r - regular with respect to µ if for every ε > 0, there exists an open set D A, D C such that h(µ(a), µ(b)) < ε, for every B C, A B D. (iv) R -regular if for every ε > 0, there are a compact set K C, K A and an open set D C, A D so that µ(b) < ε, for every B C, with B D\K. II) µ is said to be R - regular (R l - regular, R r - regular, R -regular, respectively) if every A C is a R - regular (R l - regular, R r - regular, R -regular, respectively) set with respect to µ. The reader is refereed to [7], [10] for different implications among these types of regularities in the setvalued case. Definition 2.6. Suppose C 1, C 2 are two rings so that C 1 C 2 and let µ : C 2 P f (X) be an arbitrary set multifunction. We say that C 1 is dense in C 2 with respect to µ if for every ε > 0 and every A C 2, there exists B C 1 so that B A and µ(a\b) < ε. Remark 2.7. I) B 0 is dense in B with respect to a R -regular set multifunction µ : B P f (X). Indeed, for every ε > 0 and every A B, there are a compact set K B, K A and an open set D B, A D so that µ(b) < ε, for every B B, with B D\K.

By [5], there is K 0 B 0 so that K K 0 D. Then µ(a\k 0 ) < ε, that is, B 0 is dense in B with respect to µ. II) If µ : C P 0 (X) is a fuzzy set multifunction, then µ is non-atomic if and only if for every A C, with µ(a) {0}, there exists B C, with B A, µ(b) {0} and µ(a\b) {0}. 3 Null-additive fuzzy set multifunctions In this section, we present some properties of regular null-additive fuzzy set multifunctions. By [9], one can obtain: Theorem 3.1. Let A B with µ(a) {0} and µ : B P f (X) a R l -regular null-additive fuzzy set multifunction. Then: (i) A is an atom of µ if and only if (1)! a A so that µ(a\{a}) = {0}; (ii) µ is non-atomic if and only if for every t T, µ({t}) = {0}. Theorem 3.2. Let C = B 0 (or B) and µ : C P f (X) a R-regular null-additive fuzzy set multifunction. If µ satisfies the property (2) B C, µ(b) {0} and t T, A t C s.t. t A t and e(µ(b), µ(a t )) > 0, then µ is non-atomic. Proof. Suppose that, on the contrary, there exists an atom B C of µ. Because µ is R-regular then, according to [10], it is R l -regular. Consequently, for ε = µ(b), there exists a compact set K C so that K B and h(µ(b), µ(k)) < µ(b). We observe that µ(k) {0}. Indeed, if µ(k) = {0}, then µ(b) < µ(b), which is false. According to property (2), we have that for every t K, there exists A t C so that t A t and e(µ(b), µ(a t )) > 0. Because µ is R-regular then, according to [10], it is R r -regular. In consequence, for every t K, for A t and ε = e(µ(b), µ(a t )), there exists an open set D t C so that A t D t and e(µ(d t ), µ(a t )) h(µ(d t ), µ(a t )) < < e(µ(b), µ(a t )). Since t A t and A t D t, then K D t. t K Consequently, there exists p N such that K p D t i, with t i K, for every i {1,..., p}. This implies {0} µ(k) = µ( p (D ti K)). One can easily check that there is j {1,..., p} such that µ(d tj K) {0}. Indeed, if µ(d ti K) = {0}, for every i {1,..., p}, then, by the null-null-additivity of µ, we have µ(k) = µ( p (D ti K)) = {0}, which is a contradiction. Consequently, there is j {1,..., p} such that {0} µ(d tj K) µ(d tj B). Since B is an atom of µ, µ(b) {0} and µ(d tj B) {0}, then µ(b\d tj ) = {0}, hence, the null-additivity of µ implies that µ(b) = µ(d tj B). On the other hand, because e(µ(d tj ), µ(a tj )) < e(µ(b), µ(a tj )) and e(µ(d tj B), µ(d tj )) = 0, we have e(µ(b), µ(a tj )) = e(µ(d tj B), µ(a tj )) e(µ(d tj B), µ(d tj )) + e(µ(d tj ), µ(a tj )) = = e(µ(d tj ), µ(a tj )) < e(µ(b), µ(a tj )), which is a contradiction, so µ is non-atomic. Theorem 3.3. Let µ : B P f (X) be a R- regular null-additive fuzzy set multifunction. Then µ is non-atomic if and only if (3) B B, with µ(b) {0} and t T, e(µ(b), µ({t})) > 0. Proof. Obviously, for every t T, {t} B. The Only if part is a consequence of Theorem 3.2. For the If part, suppose µ is non-atomic. Then, by Theorem 3.1, for every t T, µ({t}) = {0}, so e(µ(b), µ({t})) = µ(b) > 0. Proposition 3.4. [12-18] Let µ : C P bf (X) be a semi-convex null-additive fuzzy set multifunction. Then µ is non-atomic. Proposition 3.5. Suppose µ : B P f (X) is a R -regular null-additive fuzzy set multifunction. If µ is non-atomic on B, then µ is non-atomic on B 0. Proof. Suppose that, on the contrary, there exists an atom A B 0 for µ /B0.

Then µ(a) {0} and for every B B 0 with B A we have that µ(b) = {0} or µ(a\b) = {0}. Because A B, µ(a) {0} and µ is non-atomic on B, there is B 0 B so that B 0 A, µ(b 0 ) {0} and µ(a\b 0 ) {0}. Then µ(b 0 ) > 0 and, since B 0 is dense in B, then for ε = µ(b 0 ), there exists C 0 B 0 so that C 0 B 0 and µ(b 0 \C 0 ) < ε. Now, because C 0 B 0 and C 0 A, by the assumption we made we get that µ(c 0 ) = {0} or µ(a\c 0 ) = {0}. If µ(c 0 ) = {0}, then µ(b 0 ) = µ(b 0 \C 0 ) < µ(b 0 ), which is false. Consequently, µ(a\c 0 ) = {0}, so µ(a\b 0 ) µ(a\c 0 ) = 0, which implies µ(a\b 0 ) = {0}, which is again false. So, µ is non-atomic on B 0. Concluding remarks In this paper we have presented some properties regarding non-atomicity of fuzzy set multifunctions defined on the Baire (or Borel) δ-ring B of a Hausdorff locally compact space and taking values in P f (X), the family of nonvoid closed subsets of a real normed space X. Thus, if µ : B P f (X) is a R-regular nulladditive fuzzy set multifunction, then non-atomicity of µ is equivalent to the following condition: for every B B so that µ(b) {0} and every t T, it holds e(µ(b), µ({t})) > 0. References: [1] Aumann, R.J., Shapley, L.S. - Values of Nonatomic Games, Princeton University Press, Princeton, New Jersey, 1974. [2] Croitoru, A., Gavriluţ, A., Mastorakis, N., Gavriluţ, G. - On different types of non-additive set multifunctions, WSEAS Transactions on Mathematics, 8 (2009), 246-257. [3] Dempster, A.P. - Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Statist. 38 (1967), 325-339. [4] Denneberg, D. - Non-additive Measure and Integral, Kluwer Academic Publishers, Dordrecht/Boston/London, 1994. [5] Dinculeanu, N. - Vector Measures, Pergamon Press, Oxford, 1967. [6] Drewnowski, L. - Topological rings of sets, continuous set functions, Integration, I, II, III, Bull. Acad. Polon. Sci. Sér. Math. Astron. Phys. 20 (1972), 269-276, 277-286. [7] Gavriluţ, A. - Properties of regularity for multisubmeasures, An. Şt. Univ. Iaşi, Tomul L, s. I a, 2004, f. 2, 373-392. [8] Gavriluţ, A. - Regularity and o-continuity for multisubmeasures, An. Şt. Univ. Iaşi, Tomul L, s. I a, 2004, f. 2, 393-406. [9] Gavriluţ, A. - Non-atomicity and the Darboux property for fuzzy and non-fuzzy Borel/Baire multivalued set functions, Fuzzy Sets and Systems, Vol. 160, Issue 9, 2009, 1308-1317. [10] Gavriluţ, A. - Regularity and autocontinuity of set multifunctions, Fuzzy Sets and Systems, vol 161, issue 5 (2010), 681-693. [11] Gavriluţ, A. - Non-atomic set multifunctions, Proceedings of the 4th WSEAS International Conference on Computational Intelligence (CI 10), Bucharest, Romania, April 20-22, 2010, 74-79. [12] Gavriluţ, A., Croitoru, A. - Non-atomicity for fuzzy and non-fuzzy multivalued set functions, Fuzzy Sets and Systems, 160 (2009), 2106-2116. [13] Gavriluţ, A., Croitoru, A. - Fuzzy multisubmeasures and applications, Proceedings of the 9th WSEAS International Conference on Fuzzy Systems, [FS 08], Sofia, Bulgaria, May 2-4, 2008, 113-119. [14] Gavriluţ, A., Croitoru, A., Mastorakis, N.E. - Finitely purely (pseudo) atomic set multifunctions, WSEAS Computational Intelligence Conference, Iv. Javakhishvili Tbilisi State University, Tbilisi, Georgia, June 26-28, 2009, 103-108. [15] Gavriluţ, A., Mastorakis, N.E. - On regular multisubmeasures and their applications, WSEAS Computational Intelligence Conference, Iv. Javakhishvili Tbilisi State University, Tbilisi, Georgia, June 26-28, 2009, 109-114. [16] Hu, S. Papageorgiou, N.S. - Handbook of Multivalued Analysis, vol. I, Kluwer Acad. Publ., Dordrecht, 1997.

[17] Mastorakis, N.E. - Fuzziness in Multidimensional Systems, Computational Intelligence and Applications, WSEAS-Press, 1999, 3-11. [18] Mastorakis, N.E., Gavriluţ, A., Croitoru, A., Apreutesei, G. - On Darboux property of fuzzy multimeasures, Proceedings of the 10th WSEAS, International Conference on Fuzzy Syst. [FS 09], Prague, Czech Republic, March 23-25, 2009. [19] Narukawa, Y., Murofushi, T., Sugeno, M. - Regular fuzzy measure and representation of comonotonically additive functional, Fuzzy Sets and Systems, 112 (2000), 177-186. [20] Olejček, V. - Darboux property of regular measures, Mat. Cas. 24 (1974), no 3, 283-288. [21] Pap, E. - Regular null-additive monotone set function, Novi Sad. J. Math. 25, 2 (1995), 93-101. [22] Pap, E. - Null-additive Set Functions, Kluwer Academic Publishers, Dordrecht-Boston- London, 1995. [23] Pham, T.D., Brandl, M., Nguyen, N.D., Nguyen, T.V. - Fuzzy measure of multiple risk factors in the prediction of osteoporotic fractures, Proceedings of the 9th WSEAS International Conference on Fuzzy System [FS 08], Sofia, Bulgaria, May 2-4, 2008, 171-177. [24] Precupanu, A. - Some properties of the (B M)- regular multimeasures, An. Şt. Univ. Iaşi 34 (1988), 93-103. [25] Shafer, G. - A Mathematical Theory of Evidence, Princeton University Press, Princeton, N.J., 1976. [26] Suzuki, H. - Atoms of fuzzy measures and fuzzy integrals, Fuzzy Sets and Systems, 41 (1991), 329-342. [27] Wu, C., Bo, S. - Pseudo-atoms of fuzzy and nonfuzzy measures, Fuzzy Sets and Systems, 158 (2007), 1258-1272. [28] Zadeh, L.A. - Fuzzy Sets, Information and Control, 8 (1965), 338-353.