Measure of a fuzzy set. The -cut approach in the nite case

Fuzzy Sets and Systems 123 (2001) 93 102 www.elsevier.com/locate/fss Measure of a fuzzy set. The -cut approach in the nite case Carlo Bertoluzza a;, Margherita Solci a, Maria Luisa Capodieci b a Dpto. di Informatica e Sistemistica, Universita di Pavia, via Ferrata 1, I-27100 Pavia, Italy b UCLA (Univ. Centro-occidental Lisandro Alvarado), Barquisimeto, Venezuela Received 14 October 1998; received in revised form 19 April 2000; accepted 3 May 2000 Abstract In this paper we study the problem of constructing a Sugeno measure m on a suitable family S of fuzzy subsets of a space. More precisely, we suppose that there exists a measure m on a crisp measurable space (; S), we request that all the -cuts of the elements of S belong to the algebra S of the measurable crisp subsets and we suppose that the membership function assumes only nitely many values 1;:::; n. Under these hypotheses we construct the measure m( A) of the fuzzy set A in terms of the measure m(a i)ofits-cuts A i; i=1;:::;n, in the case where both m and m are decomposable. c 2001 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy measure; -cut; Triangular conorm; L-fuzzy sets 1. Introduction and the problem In previous works we studied the possibility of constructing the measure of a fuzzy subset of a space with nitely valued membership functions, in terms of the -levels, that is in terms of the crisp subsets A i = {! A(!)= i } (see e.g. [1]). Now, we attempt to reach the same objective starting from the -cuts (the crisp subsets A i = {! A(!) i }). In order to motivate this research we can observe, rst of all, that this last notion is more natural in a fuzzy environment than the notion of -levels. This is evident, in particular, when we examine the fuzzy sets while having in mind, as in the present case, the construction of a measure. If the membership function A assumes only a nite number of values, namely the image set is ran( A)= { 1 ;:::; n } (nite case), then both -levels and -cuts usually have a positive measure. But let us examine what happens when the image of A is the whole interval [0; 1] (continuous case). In the most natural measure space often the -levels are reduced to sets of null measure. This causes some problems when we try to extend the results obtained in the nite case to the continuous one, at least when the considered measures are not Archimedean. Corresponding author. Tel.: +39-382-5059-18; fax: +39-392-5053-73. E-mail address: carlo@ipv36.unipv.it (C. Bertoluzza). 0165-0114/01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved. PII: S 0165-0114(00)00074-9

94 C. Bertoluzza et al. / Fuzzy Sets and Systems 123 (2001) 93 102 In contrast, the -cuts maintain non-null measure even in the continuous case. This leads us to hope that the results which we obtain here (nite case) could be extended to the continuous case, even when the measure composes of an arbitrary continuous t-conorm. A dierent approach to this problem has been proposed by Grabisch et al. [2]. They generalized the previous works of Murofushi and Sugeno [3,4] providing a denition of fuzzy measure based on Sugeno and Choquet integral. Denition 1. A Sugeno measure on a measurable space (; A) isamap : A [0; 1] with the following properties: () =1; ( ) =0; A B (A)6(B): It is decomposable (see e.g. [5, pp. 27, 192; 1]) if there exists a function :[0; 1] 2 [0; 1] (composition law) such that R; S A; R S = (R S) =[(R);(S)]: (1) Theorem 1. It has been proved that is always a t-conorm and under the hypothesis of continuity it may be written in the following form: { i [ i (x)+ i (y)] if (x; y) ]a i ;b i [ 2 ; (x; y) = (2) max(x; y) otherwise; where {]a i ;b i [; i I} is a nite or countable family of open disjoint intervals; i :[a i ;b i ] R (local additive generator) is a strictly increasing function with i (a i )=0 and i is the pseudo-inverse of i dened by (t)= 1 i [min{t; i (b i )}]. i The set =[0; 1]\ i ]a i;b i [={x [0; 1] (x; x)=x} (the set of the idempotent elements) will assume great importance in the development of the present paper. It is important to remember that if x then y [0; 1] (x; y) = max(x; y): The Problem. We suppose that a decomposable Sugeno measure m is dened on a measurable space (; S). Let L = { 0 =0; 1 ;:::; n =1 i i+1 } be a nite family of values in [0; 1] such that L 1 L and let S be the family of the L-fuzzy subsets of which have all the -cuts in S. We will determine a decomposable measure m dened on S which is compatible with the measure m. The decomposability of measures of a fuzzy set has the same meaning as that of the classical one, that is A B = m( A B )= [ m( A); m( B )], whereas the compatibility condition means that m( A) is a suitable function of the measures m(a i )ofthe-cuts A i = {! A(!) i } of A m( A) =G[m(A 1 );:::;m(a n )]: (3) The second compatibility condition would be A = A is crisp m(a) =m(a) (4) (complete coherence condition). Although this condition seems to be quite natural, cases may exist where it does not hold. We justify this assessment by observing that the outcome of the crisp event A has dierent

C. Bertoluzza et al. / Fuzzy Sets and Systems 123 (2001) 93 102 95 meanings in the context of algebra S (where all the events are crisp) and in the context of family S (which contains fuzzy events beyond the crisp ones). Our problem consists of determining the most general form of function G satisfying condition (3) and possibly (4). The solution which we give here is restricted to the case where the measures m; m, which are decomposable by hypothesis, also have continuous corresponding composition laws F ; F. So they are characterized by the result of Theorem 1. We will denote by ; the sets of the idempotents, ] a i ; b i [; ]a j ;b j [ the open intervals appearing in (2), and by f i ;f j the local additive generators of the composition laws F;F, respectively. We conclude this paragraph by establishing a fundamental functional equation which follows from the fact that the measure m( A B ) may be obtained in two dierent ways: (1) rst compute m( A); m( B ) by means of the function G, then compose these measures by F: (2) rst compose m(a i );m(b i ) by means of the function F, then compose these by the function G. By setting x i = m(a i ); y i = m(b i ), the fundamental compatibility functional equation which we obtain is F[G(x 1 ;:::;x n );G(y 1 ;:::;y n )] = G[F(x 1 ;y 1 );:::;F(x n ;y n )]: (5) Moreover it is easy to recognize that, for G to dene a measure, its domain must be the set = {(x 1 ;:::x n ) x i x i+1 }, and moreover G(0;:::;0)=0; G(1;:::;1)=1; G is non-decreasing: (6) (7) (8) Denition 2. G is a fuzzy measure function (FMF) if it is continuous and satises conditions (5) (8). Denition 3. An FMF G is completely coherent if the following further condition holds: G(x;:::;x)=x; (9) which corresponds to complete coherence condition (4). 2. The fuzzy measure of a crisp set In this section we will determine the form of the fuzzy measure of a crisp subset, that is of a subset with membership function A(!) =A(!) = { 1 if! A; 0 otherwise: (10) Let us remember that all the -cuts of such a set are equal to the crisp subset A. So, if we introduce the function g(x) =G(x;:::;x); (11) we have m(a) =g[m(a)]: (12)

96 C. Bertoluzza et al. / Fuzzy Sets and Systems 123 (2001) 93 102 It is easy to recognize that the fundamental equation (5) and the properties (6) (8) determine the following conditions which characterize the function g: F[g(x);g(y)] = g[f(x; y)]; g is continuous; g is non-decreasing; g(0)=0; g(1)=1: (13) (14) Proposition 1. g()=. Proof. From Eq. (13), also remembering that, if x, then F(x; x)=x, we have F[g(x);g(x)] = g(x): This proves that g(x), that is g(). To prove that g() let us suppose that there exists z ; z= g(). Since g is surjective, there exists x ]a i ;b i [ c such that z = g(x). Letting y = x in Eq. (13), and remembering that z, but x=, we have z = g(x)= F[g(x);g(x)] = g[f(x; x)] = g[f i (2f i (x))] and by recursivity z = g[f i (nf i (x))]: Since f i (x) 0 we will have: z = lim n + g[f i (nf i (x))] = g(b i ), which contradicts our hypothesis. Remark. As an immediate consequence of this proposition we can conclude that, if I = card({]a i ;b i [}) +, then g(]a i ;b i [)=]a i ; b i [, whereas if I =+, then either g(]a i ;b i [)=]a j ; b j [ for some j, org(]a i ;b i [) is reduced to a point a. Proposition 2. A function g is the restriction of an FMF; that is g satises conditions (13) and (14); if and only if its form is given by (x) if x ; g(x) = a j if x ]a i ;b i [ and a j = b j ; (15) f j [c i f i (x)] if x ]a i ;b i [ and a j b j ; where a j = g(a i ); b j = g(a i ); c i is a positive constant and : is a non-decreasing function (with (0)=0; (1)=1) subject only to conditions which ensure the global continuity of function g. Proof. For the sake of simplicity, the proof will be given in the case where I +, so that a j =a i ; b j = b i. Nevertheless, it also holds in the general case (see [6]). : It is evident that conditions (14) hold. With respect to (13), we can observe that, if x; y does not belong to the same open interval ]a; b[ c, then (13) reduces to the identity max[g(x);g(y)] = g[max(x; y)], due to the monotonicity of g. If both x; y belong to ]a i ;b i [, then (13) is veried by the construction. : Let (x)=g = (x) be the restriction of g to the idempotent set. Conditions on function are obviously satised. Moreover if x (or y ), then compatibility condition (13) is always satised since (x) (or (y) ). If x ]a i ;b i [, then g(x) ]a i ; b i [ and therefore, by the continuity of functions g; f i ;f i, in a suitable open subset of ]a i ;b i [ ]a i ;b i [ Eq. (13) takes the form f 1 [ f i {g(x)} + f i {g(y)}] =g[fi 1 {f i (x)+f i (y)}]: (16) i

C. Bertoluzza et al. / Fuzzy Sets and Systems 123 (2001) 93 102 97 By posing u = f i (x); v= f i (y); H()= f i g f 1 (), Eq. (16) may be rewritten as i H(u + v)=h(u)+h(v): Its general solution is H(u)= f i g fi 1 (u)=c i u, from which we obtain g(x)= f i [c i f i (x)]. Finally, if x; y belong to two disjoint open intervals of c, then g(x);g(y) belong to two disjoint open intervals of c and therefore g[f(x; y)] = g[max(x; y)] = max[g(x); g(y)] = F[g(x); g(y)]. No other conditions are imposed by the compatibility equation; so the theorem is completely proved. 3. One-valued fuzzy sets In this paragraph we will determine the measure of a fuzzy set whose membership function assumes only one meaningful (non-null) value. Let r be a membership value in L and let A be a fuzzy subset of type { r if! A; A(!)= r A = (17) 0 otherwise: The fuzzy measure of A is given by m( A)=G[m(A);:::;m(A) ; 0;:::;0] def = g }{{} r [m(a)]: (18) r times It is easy to prove, using the same arguments as in Section 2, that g r ()= r = [0;g r (1)], and that the function g r has the following representation: Theorem 2. The function g r represents the measure of the one-level fuzzy set if and only if it has the following form: r (x) if x ; g r (x) = a r; j if x ]a i ;b i [ and a r; j = b r; j ; (19) f j [c r; i f i (x)] if x ]a i ;b i [ and a r; j b r; j ; where a r; j = g r (a i ); b r; j = g r (a i );c r; i is a positive constant and r : r is a non-decreasing function (with r (0)=0; r (1)61) subject only to conditions which ensure the global continuity of the function g r. Note that the function g in Section 2 coincides with the function g n. Moreover; as m( r A)6 m( s A) when r 6 s ; the functions r and the constants c r; i have to be chosen so as to guarantee the condition r 6 s g r (x)6g s (x): 4. Two-valued fuzzy sets In this section we will determine the measure of a fuzzy set whose membership function assumes only two meaningful (non-null) values r (for some r n) and n = 1. This is fundamental because it will be the basis on which we realize a recursive process which will furnish, in the end, the measure of any nitely valued fuzzy subset. The fuzzy sets we deal with in this section are a r if! B; A(!)= 1 if! C; (20) 0 otherwise;

98 C. Bertoluzza et al. / Fuzzy Sets and Systems 123 (2001) 93 102 where B; C are crisp subsets of. In this case the two meaningful -cuts of Now, let us introduce the function Gr (x; y) =G(x;:::;x;y;:::;y): }{{} r times A are A r = B C and A n = C. It is easy to recognize that, by letting x = m(a r ); y= m(a n ), we have m( A)=G r (x; y) and moreover, by denition Gr (y; y)=g(y;:::;y; y;:::;y)=g(y). The following conditions, as a consequence of (5) (8), hold: Dom(G r )= 2 = {(x; y) x y 0)}; G r is monotonic in both variables; Gr (0; 0)=0; Gr (1; 1)=1; F[G r (x; y);gr (x ;y )] = Gr [F(x; x );F(y; y )]: (21) From now on, the last relation will be indicated as compatibility equation. Let g(x); g r (x) be two functions of type (15), (19), respectively, subject to the further conditions g r (x)6 g(x), that is r (x)6(x); c r; i 6c i ; a r; j 6 a j. Then the following representation holds: Theorem 3. With the same notations as in Theorems 1 and 2; the general solution of system (21) is (a) max[g r (x);g(y)] if x or y ; (b) max[g r (x);g(y)] if x ]a k ;b k [; y ]a i ;b i [; k i; g r (a k ) a j ; Gr (x; y) = (c) f j [c r; k f k (x)+c i f i (y)] if x ]a k ;b k [; y ]a i ;b i [; k i; g r (a k )= a j ; (d) g(y) if (x; y) ]a i ;b i [ 2 ; g r (a i ) a j ; (e) f j [c r; i f i (x)+(c i c r; i )f i (y)] if (x; y) ]a i ;b i [ 2 ; g r (a i )= a j : (22) We will prove only the necessity of this condition: if Gr satises the functional system (21), then it has the representation (22). The proof of the suciency consists only in a tedious, but easy verication of a wide range of possible cases corresponding to all possible choices of the values x; y; x ;y in the last of the Eq. (21). Moreover we will consider, for the sake of simplicity, the case where I +, where a j =a i and f j = f i. The proof in the general case may be found in [6]. Case (a): Suppose that y. Then, by Proposition 1, g(y) (if x the proof is symmetric). Thus, the compatibility equation may be written as G r (x; y) =G r [max(x; y); max(0;y)] = max[g r (x; 0);G r (y; y)] = max[g r (x);g(y)]: Cases (b, c): In these cases F(x; y) = max(x; y). Thus, remembering that x y and by using the compatibility equation, we can write G r (x; y) =G r [max(x; y); max(0;y)] = F[g r (x);g(y)]: The proof can be completed by observing that the form (2) of the function F is max when the arguments do not belong to the same interval ] a; b[ as happens in case (b). On the other hand, F is f [ ] when, as in case (c), the arguments belong to the same interval. Moreover, we recall that the form of the functions g r ;g is given by Propositions 1 and 2.

C. Bertoluzza et al. / Fuzzy Sets and Systems 123 (2001) 93 102 99 Fig. 1. Cases (d, e): For all (x; y) in]a i ;b i [ ]a i ;b i [ we have a i = G r (a i ;a i )6G r (x; y)6g(b i ;b i )= b i. So in this case the compatibility equation becomes f [ f i Gr (x; y); f i Gr (x ;y )] = Gr [f i {f i (x)+f i (x )};f i {f i (y)+f i (y )}]; i which in a suitable open subset of ]a i ;b i [ ]a i ;b i [ ]a i ;b i [ ]a i ;b i [ may be written as H(u; v)+h(u ;v )=H(u + u ;v+ v ); (23) where u = f i (x); v= f i (y); u = f i (x ); v = f i (y );H= f i G r f i. The general continuous solution of Eq. (23) is the family of bilinear homogeneous functions (H(u; v)=au + Bv). Consequently, we have Gr (x; y) = f i [Af i (x)+bf i (y)]: (24) By posing x = y in (24) we easily recognize that A + B = c i : (25) Moreover, let us observe that Gr (x; y) Gr (x; a i ) = max[gr (x; 0);Gr (a i ;a i )] = max[gr (x; 0); a i ]. On the other hand, according to (24), G r (x; a i )= f i [Af i (x)]. By comparing the two expressions of G r (x; a i ) we can conclude that If Gr (x; 0) a i, then f i (a i )=Af(x) and, since f i (a i ) = 0 we have to conclude that A = 0, and from (25), B = c i. This happens when Gr (a i ; 0) a i, thus case (d) is proved. If Gr (x; 0) a i then f i [Af i (x)] = Gr (x; 0) = f i [c r; i f i (x)], and therefore A = c r; i, and from (25), B = c i c r; i. This happens when Gr (a i ; 0) = a i, thus case (e) is proved. 5. The measure of a fuzzy set Now, by using the result of the previous section, we can install an iterative procedure which will give us the general form of the fuzzy measure function G(x 1 ;:::;x n ). Intuitively speaking, the idea is the following. Starting from a fuzzy subset A with membership values in L = { 0 =0; 1 ;:::; n =1 i i+1 }, we construct a sequence { A (i) i =1;:::;n} of fuzzy sets where A (n) = A, and A (i) is obtained from A (i+1) by suppression of two -cuts and insertion of a new -cut, which is equivalent to the suppressed ones from the point of view of the measure. More precisely, what we try to impose is that the fuzzy measures of each A (i) are equal to the fuzzy measure of A. We stop the process when i = 1, that is when we have obtained a one-level fuzzy set.

100 C. Bertoluzza et al. / Fuzzy Sets and Systems 123 (2001) 93 102 Then we apply the result of Section 2 to obtain the measure of A. This could be interpreted as a procedure that (see Fig. 1). substitutes -cut A n by a suitable subset A, destroys A n 1, thus eliminating the value n 1 in L, completes the -cut A n 2 by setting A (n 1) (!)= n 2 ;! A n 1 A. Thus, we constructed a new fuzzy subset A (n 1) which has membership values in L n 1 = L { n 1 }. The idea is to organize the construction of A in such a way that m( A (n 1) )= m( A). Then the procedure restarts, now with n = n 1, and proceeds with diminishing n at each step, until n = 1. Formally, we proceed as follows. We start by supposing that G(x 1 ;:::;x n )=G[x 1 ;:::;x n 2 ; n (x n 1 ;x n )]; (26) that is we suppose that the function G depends on its last two arguments only by means of a suitable real function of x n 1 ;x n. Although it seems natural in the procedure which constructs the measure level by level, we are really making a new restriction on the class of allowable measures. It looks like a kind of branching property. It is easy to check that the domain of n is the set 2 = {(x; y) x y 0}; y6 n (x; y)6x: (27) (28) By using expression (26) we recognize that the rst and second members of the compatibility equation (5) have the following form: F[G(x 1 ;:::;x n );G(y 1 ;:::;y n )] = F[G{x 1 ;:::;x n 2 ; n (x n 1 ;x n )};G{y 1 ;:::;y n 2 ; n (y n 1 ;y n )}] = G[F(x 1 ;y 1 );:::;F(x n 2 ;y n 2 ) n {F(x n 1 ;y n 1 );F(x n ;y n )}]; (29) G[F(x 1 ;y 1 );:::;F(x n ;y n )] = G[F(x 1 ;y 1 );:::;F(x n 2 ;y n 2 ); n (F(x n 1 ;y n 1 );F(x n ;y n ))]: (30) By comparing these two expressions we can conclude that the compatibility equation is satised if (and only if in the case of strict monotonicity) F[ n (x n 1 ;x n ); n (y n 1 ;y n )] = n [F(x n 1 ;y n 1 );F(x n ;y n )]: (31) From this equation we can easily deduce that n () : (32) It suces in (31) to set x n 1 = y n 1 = a; x n = y n = b and suppose that a; b. On the other hand, the function G(x 1 ;:::;x n )=G[x 1 ;:::;x n 2 ; n (x n 1 ;x n )] is an FMF. This implies that n is monotonic non-decreasing; n is continuous: (33) Eqs. (31) (33) show that the function n satises all the conditions on the function G r given in Section 4, and therefore it can be determined by the procedure in the same section. Its general form is given by formula

C. Bertoluzza et al. / Fuzzy Sets and Systems 123 (2001) 93 102 101 (22) with two slight dierences: (i) f j ; a j have to be substituted by f j ;a j, since in the actual compatibility equation (31) F has been changed to F, (ii) g(x) becomes the identity, since now, from (28), n (x; x)=x. So the form of n is (a) max[h n (x);y] if x or y ; (b) max[h n (x);y] if x ]a k ;b k [; y ]a i ;b i [; k i; h n (a k ) a j ; n (x; y) = (c) f j [c n; k f k (x)+c i f i (y)] if x ]a k ;b k [; y ]a i ;b i [; k i; h n (a k )=a j ; (d) y if (x; y) ]a i ;b i [ 2 ; h n (a i ) a j ; (e) f j [c n; i f i (x)+(c i c n; i )f i (y)] if (x; y) ]a i ;b i [ 2 ; h n (a i )=a j ; (34) where h n (x)= n (x; 0). Note that what we did here has been to transform the computation of the measure of a fuzzy set with n allowable membership values ( 1 ;:::; n 1 ; 1) into the computation of the measure of an equivalent fuzzy subset with n 1 membership values ( 1 ;:::; n 2 ; 1), by using the function n. The iteration (n 1 times) of this operation leads us to an equivalent set with only one membership value = 1, that is a crisp subset. The fuzzy measure of this subset can be obtained by means of the function g in Section 2. So, by setting x i = m(a i ), we nally obtain m(ã) =g( 2 (x 1 ; 3 (x 2 ; 4 (x 3 ;::: n (x n 1 ;x n ) :::)))): (35) 6. Two examples In the examples we present here the set of idempotents is =[0;a] [b; 1], and contains two particular important subcases: b = 1, that is =[0;a] {1}. In this case the measures less than a are negligible in the sense that as soon as there exists a measure m(a i ) greater than a, the -cuts A i with m(a i )6a are discarded. a = 0, that is = {0} [b; 1]. In this case the measures greater than b are dominant in the sense that as soon as there exists a measure m(a i ) greater than b, the -cuts A i with m(a i )6b are discarded and the global measure becomes max{m(a i )}. Example 1. We can choose the same function G to construct all the two level s set measures, that is G i = G; i with, G(a; 0) = a; G(b; 0) = b and G(1; 0)=(b +1)=2. Let us select the additive generators of m; m in ]a; b[ equal to log[(b x)=(b a)] and x, respectively. A choice of G(x; 0) compatible with the results of Proposition 5 is x if x6a; (b x)c G(x; 0) = b (b a) c 1 if a x b; x + b if x b: 2 (36)

102 C. Bertoluzza et al. / Fuzzy Sets and Systems 123 (2001) 93 102 The result of Propositions (8), (9) assigns to the function G the following form: x if (x; y) [0;a] [0;a]; b +1 if x [b; 1] [0;a]; 2 b (b x) c (b y) 1 c if x ]a; b[ ]a; b[; G(x; y) = (b x)c b (b a) 1 c if x ]a; b[ [0;a]; x + b if x [b; 1] [b; 1]: 2 (37) Example 2. We choose G(a; 0) = a; G(b; 0) = b; G(1; 0) = 1 and G 2 (x; 0) = { x if x ; f [c 2 f(x)] if a x b; where f is any additive generator, and c 2 0. From Propositions (8), (9) and formula (27) we obtain 2 (x) =x if x ; G(x; y) = f [c 2 f(x)] if a x b; y ; f [c 2 f(x)+(1 c 2 )f(y)] if (x; y) ]a; b[ 2 : (38) (39) In order to obtain the form of the measure of a multi-level fuzzy subset we assume that all the maps Gi have the same form of G, possibly with dierent constants c i, and i (x)=x. The general form of measure m is obtained by means of the following function G(x 1 ;:::;x n ): g(x 1 ) [ if x 1 ; G(x 1 ;:::;x n )= k ] i g f (1 c (40) j)c i+1 f(x i ) ; otherwise: i=1 j=2 where k = max{ j x j ]a; b[}. Moreover, c n+1 = 1 by denition. References [1] C. Bertoluzza, A. Bodini, Measures of a fuzzy set based on an arbitrary continuous t-conorm: the nite case, Proc. FUZZ-IEEE 96, New Orleans, September 8 11, 1996. [2] M. Grabisch, T. Murofushi, M. Sugeno, Fuzzy measures of fuzzy events dened by fuzzy integrals, Fuzzy Sets and Systems 50 (1992) 293 313. [3] T. Murofushi, M. Sugeno, Fuzzy t-conorm integral with respect to fuzzy measures: generalization of Sugeno integral and Choquet integral, Fuzzy Sets and Systems 42 (1991) 57 71. [4] T. Murofushi, M. Sugeno, A theory of fuzzy measures. Representation, the Choquet integral and null sets, J. Math. Anal. Appl. 159 (2) (1991) 532 549. [5] E. Pap, Null Additive Set Functions, Kluwer Academic Publishers, Dordrecht, 1995. [6] M. Solci, Misura di un insieme sfumato tramite -sezioni, Tesi di Laurea, Anno Accademico 1997=98.