- Email: [email protected]
Range of finitely additive fuzzy measures
FU2ZY
sets and systems ELSEVIER
Fuzzy Sets and Systems 89 (1997) 231-241
Range of finitely additive fuzzy measures A n n a A v a l l o n e a,*, G i u s e p p i n a B a r b i e r i b a Dipartimento di Matematica, Universita" della Basilicata, Via N. Sauro, 85, 85100 Potenza, Italy b Dipartimento di Matematica, Universita' di Udine, Via delle Scienze, 206, 33100 Udine, Italy
Received August 1995; revised March 1996
Abstract We extend Liapunov theorem to R~-valued finitely additive measures on clans of fuzzy sets. Moreover, we study the convexity o f the range of B-convex-valued measures. (~) 1997 Elsevier Science B.V. Keywords. Measure theory; Fuzzy measures; Range
O. Introduction
In this note, we study some properties of the range of finitely additive fuzzy measures on clans (for a study of fuzzy measures, see for example [7], where finitely additive fuzzy measures are called T~-valuations). Precisely, in the first part, we extend the classical Liapunov theorem concerning the convexity of the range of nonatomic R'-valued measures. This theorem has been proved for a-additive measures on aalgebras in [13] and then in [11]; recently, it has been independently proved for finitely additive measures on algebras with the interpolation property in [8, 14, 1]. Moreover, this result has been proved also for multimeasures on a-algebras (see [2,4]), for modular functions on complemented lattices (see [3]) and for a-additive fuzzy measures on tribes (see [5, 6]). In the second part, we generalize a result of [12] concerning the convexity of the closure of the range of measures with values in B-convex spaces (for a study of these spaces, see for example [9]). * Corresponding author. 0165-0114/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved PH S01 65-011 4( 96 )00099-1
Finally, we conclude with a remark concerning the relative weak compactness of the range of fuzzy measures with values in complete locally convex topological linear spaces.
1. Fuzzy measures
The aim of this section is to give a description of some properties of fuzzy measures. Let X ~ 0 a set. Notation. If f , g E [0, 1] x, we denote by f V g and f A g the functions defined, respectively, by
f V g(x) = m a x { f (x), g(x)}, f A g(x) = min{f(x),g(x)},
x E X,
and by 0 and 1 the functions defined, respectively, by f ( x ) = 0 and f ( x ) = 1 for every x E X. If {f~ : ~ E A} is an increasing (resp. decreasing) net in [0, 1]x and f ( x ) = sup~f~(x) (resp. f ( x ) = inf,(x)) for x E X, then we write f~ T f (resp. f~ ~.f). Moreover, we denote by N and R, respectively, the set of natural numbers and of real numbers.
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Definition 1.1. Let .%/~_ [0, 1] x. ,.%/is called clan if (1) 1 c ,%/. (2) I f f , g ~ , % / , t h e n ( f g) VOE~%/. ~%/is called tribe if,%/is a clan and the following property holds: for every sequence { f , } in "%/,(~,eN f , ) A 1 (: ,%/. Proposition 1.2. Let ~%/C [0, 1] x. Then ,%/is a clan if and only if the following conditions hold: (1) I f f C ,%/, then 1 - f ~ ,%/. (2) 0 ~ ,%/. (3) I f f , g ~ ~%/, then ( f + g - 1 ) V 0 ~ ~%/. Proposition 1.3. Let ,%/ be a clan. Then ~%/ is a tribe if and only iJl Jot ever), sequence {f~} in ~%/, [1 - ~,,~l(l - f,)] V 0 ~ ,%/. Proof. We {[1 -- E n Z l ( 1
u s e [~n%l(1 -- f~)] i 1 = - - f n ) ] V O} and Proposition 1.2.
1 --
Definition 1.7. Let ,%/ be a clan and ( G , + ) a semigroup. A function tt : ,%/--~ G is called (fuzzy) measure if the following properties hold: (1) ~(0) = O. (2) I f f , g E ,%/and f + g < ~ l , then # ( f + g) --
~(,1)
+ ~z(g). If G is a topological semigroup, then a measure /~ : ~%/ ---+ G is called a-additive if, for every seoo %/ cx~ quence {f~} E ~%/, with Y'~,,=I ,fn C~ , # ( ~ , = 1 J ~ , ) =
Proposition 1.8. Let ( G, +) be a group, ~%/a clan and IX : ,%/ ---+ G a function such that t~(0) - O. Then # is a measure if and only if, Jor every f , g ~ ,%/,
hi(f+
g - l) v o] + h i ( f +
g) A 1] = n ( f ) + ~ , ( g ) .
Proof. Suppose that # is a measure. Since g = [ ( f ÷ g - - 1) V 0] + [g A (l -- f ) ] a n d ( f + g ) A 1 = f + [g A (1 -- f ) ] , then
Remark. By Propositions 1.2 and 1.3, the definitions of clan and tribe are equivalent, respectively, to the definitions of Too-clan and Too-tribe of [7].
~[(f+
Proposition 1.4. Let ,%/be a clan. Then: n (1) I f f l . . . . . f~ E ,%/, then ( ~ i = l J})A 1 C °~/. (2) ff" f ,g 6 ~%/, then f V g 6 , % / a n d . f l A g 6 ~4.
from which we obtain the desired equality. The converse is trivial. []
Proofl (1) The assertion is true for n = 2, since, by (1.2),
Remark. By Proposition 1.8, the definition of measure, for G = R, is equivalent to the definition of Toovaluation of [7].
g - 1) v o] - ~ ( g ) - ~ [ g A (1 - f ) ] ,
kz[(f + g) A 1] = / ~ ( f )
+ / ~ [ g A (1 -- f ) ] ,
(.fi + .f2)n 1 =l-{[(l-j])+(1-j))-l]VO}E,%/. Then ( 1 ) follows by induction, because
(2) f A g = f - [ ( J - g ) V 0 ] E,%/andfVg= g + [ ( J - g) V 0] C ~ by Proposition 1.2. [] Corollary 1.5. Every clan is a lattice (with respect to the pointwise order).
Proposition 1.9. Let (G, + ) be a group, ¢%/a clan and IZ : , % / ~ G a measure. Then the following properties hold: (1) I f f , g ~ °%/andf<~ g, then I ~ ( g - f ) = P(g)
~(f). (2) f i G = R and ~ 9 0 , then t~ is monotone. (3) # is modular, i. e., Jbr every f , g E ~/, # ( f V g) + ~ ( / A g) = ~(.1) + ~(g). (4) if.t] . . . . . f , C ,%/ and./) + ... + J~,~< 1, then n n ~ ( ~ i : ~ f , ) = E , : , t'(.~). (5) I f G -- R and lt >10, then/~ is subadditive, i.e., /~[(~7-1 ~ ) A 1]~< ~~i~_l #(J}) fi)r every f , . . . . . f~ in.%/.
Proposition 1.6 (Butnariu and Klement [7, Remark 2.8]). Every tribe is a a-complete lattice (with respect to the pointwise order).
ProoL (1) Follows from g = f + (g - f ) . (2) Follows from (1).
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(3) S i n c e f = f A g + [ ( f - g ) [ ( f - g) V 0] + g, we obtain p(f) = p(fA
g) + / l [ ( f
V0]andfVg=
-- g) V 0],
gn E d and E k = l Ok = f n ~ 1. Therefore, by Proposition 1.9(4), ~(f) = ~
/ l ( J V g) = p [ ( f - g) V 0] + p(g), from which (3) follows. (4) Follows by induction. (5) By Proposition 1.8, (5) is true for n = 2. Because of (*) in Proposition 1.4, we obtain
Then we use induction.
~(g~) = lim~
[]
(2) ~ (3): Let fn, f E • such that fn ,[ f . Set gn = 1 - f~ and g = 1 - f . Then gn, g E d and g~ T gTherefore, by Proposition 1.9( 1 ), #(1)- #(f)=
# ( g ) = lim/~(gn)---/~(1)- lim #(f~). t/
t/
0 = lim #(gn) = # ( f )
/~-(J') = sup{-#(g) : g E ~,g<~J'} and
- lira ~2~/~(fk)
n k=l
= #(f) - ~
/ ~ + ( f ) = s u p { p ( g ) : g E ,J,g<~ f },
p-(f),J'
= lim~(f~).
k=l
(3) ~ (4): Trivial. (4) ~ (1): Let f , , f E ,~¢ such that f = ~n~_l J~. ii Set g, = f - E k = l fk- Then gn E ~4 and g, I 0. Therefore, by Proposition 1.9(4).
Definition 1.10. Let ,~/be a clan and p : ~ ' --* R a measure. The functions defined by
]#l(f) = P+(f)+
gk
n=l
It(fn) •
[]
n=l
Remark. By Proposition 1.12, the definition of aadditive measure, for G = R, is equivalent to the definition of Too-measure of [7].
E ~/
are called, respectively, positive variation, negative variation and total variation of #. Proposition 1.11 (Butnariu and Klement [7, Propositions 9.3, 9.4 and L e m m a 10.1]). Let d be a clan and p : , ~ --+ R a measure. Then the following properties hold: (1) # + , # - and I#l are measures. (2) # is bounded f f p+ and t~- are bounded iff I/tl is bounded (3) I f p is bounded, then I~ = P+ - # - .
Proposition 1.13. Let ~¢ be a tribe and t~ :su¢ ---+R a a-additive measure, with It >~O. Then # is a-subadditive, i.e. f o r every sequence {fn} in ~1, # [ ( ~ , % 1 f , ) A 1] ~< ~ , % 1 / ~ ( f , ) . Proofl Let {f~} be a sequence in ~/ and set gn = n oo ~ k = l fk and f = ~ n = l fn- Since limn(g~A 1 ) = f A 1, by Propositions 1.9(5) and 1.12, we obtain
~< lim ~-~ # ( f k ) Proposition 1.12. Let ~¢ be a clan, G a topological group and 1~ : ,~ -+ G a measure. Then the following conditions are equivalent: ( 1 ) p is a-additive. (2) f ~ T f , with f n , f E ~ ¢ , ~ / ~ ( f ) : limn #(fn). (3) f , J~f, with f , , f E ~ / , ~ p ( f ) : limn p ( f , ) . (4) f n i O , with f , E ~ ' , ~ lim, # ( f , ) = 0. Proof. (1) ~ (2): Let f . , f E ~ such that fn T f . Set g j = Ji and, for n~>2, gn = f n - f ~ - l . Then
k
1
= ~ #(A).
[]
n=l
2. Nonatomic fuzzy measures In this section, we study some properties of nonatomic fuzzy measures, which will be useful in the next sections.
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G. B a r b i e r i / F u z z y
S e t s a n d S y s t e m s 8 9 ( 1 9 9 7 ) 231 241
Let o~¢ be a clan, (G, + ) a topological group and IX : , ~ ~ G a measure. In the following, i f G is seminormed, we set, for f E ,~,
fi(f)
= sup{Nix(g)ll : g ~
,rd, g4 f } .
Definition 2.1. IX is called n o n a t o m i c (or continuous) if, for every 0-neighbourhood U in G, there exist f l . . . . . f , in ,~¢ such that f l + " " + f ~ = 1 and Ix(g) E U if g E ~¢ and there exists i ~
Definition 2.2. IX is called chained if, for every 0neighbourhood U f ~< g, there exist f 0 ~< f l ~ < ~ < f n 0, tP ~ .~/,0~<~
in G and for every f , g ~ se', f l . . . . . ];_~ in s¢ such that = g and I X ( 6 ) - Ix(q5) ~ and there exists i~
with f = U if that
.D-, ~
Proof. (2) =:> (1): Trivial. (1) ==> (2): Let U be a 0-neighbourhood in G and let x, y E L such that x ~~
r i A x~
1 ~ Si A x i
1 ~ Zi
and 2 is modular, we have 2 ( r ) + 2(zi) = ) . ( Y i ) -~- ).(ri) = 2 ( y i ) + ).(zi-j ) + 2(ri V xi 1) - ),(xi l) and
1
from which 2(s) - 2@) = 2(si V x i - 1 ) - .;t(ri V x i - i ).
Since x i - i ~ ri V x i - 1 ~ , ~ ( s ) - ) . ( r ) E U. []
Si V x i
1 ~ X i , we obtain that
Proposition 2.4. The Jollowing conditions are equivalent: (1) For every f E ~1 a n d f o r every O-neighbourh o o d U in G, there exist ,['1. . . . . fn in ,~¢ such that ,[l + " " + fn = f a n d ix(g) C U i f g E ~¢ and there exists i <~n such that g <~fi. (2) IX is" nonatomic. (3) IX is chained
Proof. (1) ~ (2): Trivial (2) =~ (3): Let U be a 0-neighbourhood in G and choose .fl . . . . . . /~ in <~_¢such that f l + "'" + fn = 1 and Ix(g) ~ U i f g E . ~ and there exists i<~n such that g~l, gi = ~ i = 1 .])" Then gi C ,~' for every i E {0 . . . . . n} and 0 = g 0 ~ < g l ~ < . . - ~ < g n = 1. Moreover, i f 0 , ~ E ~ and there exists i ~< n such that g i - l ~ if) ~ if/~ gi, then - 0 ~ - ~ and ~b - q5 ~
- h ) E U.
r~.
Proposition 2.5. Suppose G s e m i n o r m e d Then: (1) I f f~ . . . . . J'~ E ,~, then ~ [ ( ~ / n 1 f i ) / ~ 1]~< n ~i=1 ~ (fi)" (2) I f IX is" nonatomic, then # is" bounded.
Proof. (1) First suppose n = 2. Let g E o~¢ such that g~<(Ji + J2) A 1 and set h = .fl A g, k = (9 - f l ) V 0. Hence h , k E , ~ ' , h < ~ f l , k < ~ f 2 and g = h + k. Therefore,
2(s) + ;,(z~) = 2(y~) + 2(s,) 2(yi)
A- 2 ( Z i - - I ) -4- 2 ( S i V x i 1 ) - - . ~ ( x i
1),
tl~(g)ll ~< It~(h)ll + II~(k)ll ~<~(fl ) + ~(f2),
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235
Definition 2.9. Let G = R. I f f E 5u¢, we say that f is a p - atom if p ( f ) # 0 and, i f g E ~¢ and g<~f,
from which fi[(fl + f 2 ) /~ 1] <~fi(f~ ) + fi(f2). Then, using (*) in Proposition 1.4, (1) follows by induction. (2) Let f~ . . . . . f~ be in ,~¢ such that f l + "'" + f~ = 1 and f i ( f l ) < 1 for every i<.n. Then, by (1), for every f ~ ,~¢,[[#(f)[[ ~
then either # ( g ) = 0 or # ( f - g) = 0. W e say that # is atomless if there are not #-atoms. I f G = R n, # is called atomless if each component o f # is atomless.
L e m m a 2.6. Suppose G = R. Then, if f C , ~ ¢ , f i ( f ) ~< l# I ( f ) ~<2 f i ( f ) .
Proposition 2.10. I f G = R, then # is atomless /fflpl & atomless.
Proof. If g E ,~¢ and g<~f, then I # ( g ) t ~ l p l ( f ) , from which we obtain the first inequality. The other inequality follows from # - ( f ) < . . . f i ( f ) and
Proof. W e prove that f c ~¢ is a p - a t o m iff f is a
p+(f)<.fi(f).
[]
Proposition 2.7. Suppose G = R. Then the following conditions are equivalent: ( 1 ) # is nonatomic. (2) Ipl is nonatomic. (3) # + , # are nonatomic. Proof. (1) =~ (2): Let e > 0. By (1), we can choose f l . . . . . fn in .~¢ such that f l + ' " + f~ = 1 and f i ( . ~ ) < ~;/2 for every j-..< n. By L e m m a 2.6, we obtain I P ] ( ~ ) < e for e v e r y j ~ < n . (2) ~ (3): Trivial. (3) =:> (1): Let e > 0 and f l . . . . . fn in ~¢ such that f l + ' " + f ~ = l and p + ( ~ ) < e/2 f o r i < ~ n . By Proposition 2.4, for every j~< n, we can choose Si glj . . . . . g~jj in ,~' such that ~ h = l ghj = J ) and p - ( g h j ) < e/2 for every h <<.sj. Then ~ j = l ~ h s, = l ghj = f and # + ( g h j ) - - . < # + ( ~ ) < e/2. By L e m m a 2.6, we obtain fi(ghj) < ~: for every h<~sj a n d j ~ < n . []
Proposition 2.8. Suppose G = R ~. Then p & nonatomic if and only if each component o f # is nonatomic. Proof. Let p = (#! . . . . . Pn). Suppose that, for every i ~< n, #i is nonatomic. We use induction. Let e > 0. By assumption, p ' = (p 1. . . . . # n- 1) is nonatomic. Then, by L e m m a 2.6, we can choose f l . . . . . fm in ~ such that f l + ' " + J m = 1 and I # i l ( ~ ) < e for i<.n - 1 and j ~
I# I-atom, Suppose f is a [#[-atom. Then we can choose 9 E ,s¢ such the g < ~ f and # ( 9 ) ¢ 0. Hence, by assumption, ] P l ( f - O ) = 0 and therefore # ( g ) = # ( f ) # O. Then f is a p-atom. N o w suppose f is a #-atom. It is sufficient to prove that, i f g E ~¢,g<<.f and # ( g ) = 0, then [pl(g) = 0. For this, we prove that, if h E ~¢ and h ~
Lemma 2.11. Suppose G = R and # >~0. Suppose that, Jor every f E ~¢ and c¢¢ R such that 0 < < p ( f ) , there exists 9 E ~¢ such that g<<.f and #(g) = ~. Then, f o r every f E ~4 a n d m E N, there exist f l . . . . . fm in z¢ such that f l + "'" + fm = f and # ( f i ) = ( 1 / m ) # ( f ) f o r every i <~m. Proof. Let f E ~¢ and m E N, with m>~2. By assumption, we can inductively choose f l . . . . . fro-1 in d such that f l ~ < f , f i ~ < f ~-{fj for i~>2 and # ( f i ) = ( 1 / m ) p ( f ) for every i<.m - 1. Then, m--I m set fm = f - Y ~ i = l fi, fm C d , ~ i = l j ) = f and
#(fro) = ( 1 / m ) p ( f ) .
[]
Proposition 2.12. Suppose G = R". Then the followin 9 conditions hold: (1) I f # is nonatomic, then # is atomless. (2) I f # is a-additive and ~ is a tribe, then p is nonatomic iff # is atomless. Proof. (1) By (2.7), (2.8) and (2.10), we can suppose G = R and p >/0. Let f C ~¢, with p ( f ) :~ 0. Choose e such that 0 < e < p ( f ) and f l . . . . . fm in ~¢ such that f l + " " + f m = f and #(J~) < e for every i ~ m . Since 0 < # ( f ) = ~']~iml#(.1~), there
A. Avallone, G. Barbieri/Fuzzy Sets and Systems 89 (1997) 231~41
236
exists, j~< m such that # ( f j )
-#(j'))
> 0 and # ( f - f j ) >
> 0.
(2) Suppose # is atomless. Like in (1), we can suppose G = R and # >/0. By [6, Theorem 2], the assumption o f L e m m a 2.11 is satisfied. Then, by L e m m a 2.11, # is nonatomic. []
3. Range of fuzzymeasures 3.1. Liapunov theorem In this section, we extend to fuzzy measures the classical Liapunov theorem concerning the convexity o f the range o f nonatomic R"-valued measures. We follow the idea o f [1 1]. Let o~¢ be a clan, n E N and # : ,~¢ --~ R ~ a measure, with # = (#~ . . . . . #~). I f f ~ , ~ ' , we set [0, f ] = {g E ,~¢ : g ~ < / }
Definition 3.1.1. We say that ~ ' has the interpolation property if, for every sequences {f~}, {g~} in ~¢, with f,<~J~+~ <~g,+~ <~g, for every n ~ N, there exists f E ~¢ff such that f~ ~
( 1 ) It is semiconvex. (2) For every f E ~¢ and 2 E [0,1], there exists f;. E zff such that f;<~J', #(f;~) = 2 # ( f ) and f;<~f~, / f 2 --.<2'. (3) For every f E ~¢, # ( [ 0 , f ] ) is" convex. Proof. (1) =~ (2): Let f E ~ ' . By induction, we can obtain, for every dyadic rational k C [0, 1], fk c ~q/ such that fk ~0 for every i<~n, we obtain (2) for every 2 E [0, 1]. (2) ~ (3): Let f , g , h E ~u¢, with g,h E [0, f ] and ), E [0, 1]. By L e m m a 3.1.4, there exist q~, ~p, k E , ~ such t h a t q ~ + ~ + k ~ < f , g=q~+kandh=~+k. S e t y = d p ; + ~ p l ; +k. Yhen T~< f , y E ~ ' and # ( 7 ) = 2 # ( g ) + (1 - 2 ) # @ ) . (3) ~ (1): Trivial. [] Proposition 3.1.6. Suppose that ~¢ has the interpolation property and # is" nonatomic, with #i ~ 0 Jbr every i <~n. Then, set zi = ~ - i #k Jor every i <~n, = (zl . . . . . z , ) is semiconvex., Proof. If n = 1, we can apply Theorem 3.1.3. Then we use induction. Suppose that z = (z~ . . . . . zk) is semiconvex and set n = k + 1. Let f C ~¢. By assumption, we can choose g E o~¢ such that g ~
for every i~< k.
Moreover, by Proposition 3.1.5, for every 2 E [0, 1], we can choose g;~ E ,~¢ such that g;. <~g, ri(g;.) = A'Ci((] ) for every i<~k and g;<~ g;, if2~< 2'. N o w we remark that, if 21 ~<22, then 0 ~ r k + l ( g 2 : ) - zk+l(g~,)<~rk(g& -- g2,) =(22
Lemma 3.1.4. Let f , g , h E ~¢, with g<~ f andh<~ f . Then there exist O , O , k ~ ~ such that 0 + qJ + k <~f ,g = qS + k and h = tp + k. Proof. Set k = g A h , ( h - g ) V 0. []
~b = ( g - h )
V 0 and ~9 =
Proposition 3.1.5. Suppose that ~ has the interpolation property and #i ~ 0 f o r every i <. n. Then the following conditions are equivalent:
(*)
- 21)¢k(g),
from which we obtain that the function 2 E [0, 1] z~+l(g;.) E R is continuous. In a similar way, we obtain the continuity o f the function 2 E [0, 1] ---+ zk+~[(f -- g)l-;~] E R and therefore, set h ;~ = g;. +(f-gh-;., the function ~ ( 2 ) = Zk+l(h ;~) is continuous on [0, 1]. By (*), the assertion is true if rk+l(g) = ½ r k + l ( f ) or r k + l ( f g) = ~1 r * + l ( ] ) - Then we can suppose Zk+l(g) < ½zk+l(f). Since ~(0) = r k + l ( f - g ) , a(1) = zk+i (g) and a is continuous, there
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exists 20 E [0, 1] such that rk+l(h ;-°) = Moreover, if 1 ~< i ~
l'Ck+l(f).
ri(h ;'° ) = 2ori(g) + (1 - 2o) zi ( f - g)
= lz/(f).
[]
T h e o r e m 3.1.7. S u p p o s e that ~¢ has the interpolation p r o p e r t y and ff is' nonatomic. Then, f o r every f c o~, #([0, f ] ) is convex. Proof. Let f E ~ ' . First suppose ]2i ~0 for every i<<.n and let z be like in Proposition 3.1.6. By Propositions 3.1.5 and 3.1.6, r([0, f ] ) is convex. Therefore, #([0, f ] ) is convex too, because, if we set, for t = (tl . . . . . tn) E R n, T ( t ) = (tl - t2 . . . . . tn--1 -t,, t,), then T is linear and p = T o z. Now we remove the assumption ]Ai ~0 for every i~< n. Set 2 = (Pl+ ,Pl , ' " , / ~ n+, / 4 - ) " By Propositions 2.7 and 2.8, 2 is nonatomic. Then ) ~ ( [ 0 , f ] ) is convex. Hence, p([0, f ] ) is convex too, because, if we set, for t = ( h . . . . . t2n) E R 2~, T ( t ) = (tl - t2,t2 - t3 . . . . . t2n-l - t2~), then T is linear and IL =
To)..
therefore ( f l ( Y ) . . . . . f . ( y ) ) E v([0, f ] ) . Then there exists g E o~¢ such that g < . f and J } ( y ) = f i ( 2 ( g ) ) f o r e v e r y i ~ n . Hence y E 2([0, f ] ) w. [] 3.2. R a n g e o f f u z z y measures with values in B - c o n v e x spaces
We shall extend to fuzzy measures a result o f [12] concerning the range o f measures with values in Bconvex spaces. Let E be a normed linear space, d a clan and p : ~¢ ~ E a measure. First we prove that there exists a measure #* : .~---~ [0, + o c ] such that, for every f E , d , [[P(f)l[ ~< Y ( f ) . This is a tool in the p r o o f o f Theorem 3.2.9. Definition 3.2.1. If f E s~', we set
n
}
z
"=
R e m a r k . By [7, Proposition 15.10], if E = R, p* is the total variation o f p. L e m m a 3.2.2. If f E ~4, then if(f)
= sup { £" ,l~(Ji) / = - 1~(J~-,)l
C o r o l l a r y 3.1.8. S u p p o s e that A has the interpolation property, E is a locally c o n v e x H a u s d o r f f topological linear space a n d ), : ~4 -+ E is" a n o n a t o m i c measure. Then, J o t ever)' f E ~ , the w e a k closure o f ) , ( [ 0 , f ] ) is" equal to the closed convex hull
oj;.([o,
~f~ f
and
[]
R e m a r k . Theorem 3.1.7 is a generalization o f Theorem 1 o f [6] about convexity, because every tribe has the interpolation property (since it is a acomplete lattice by Proposition 1.6) and because o f Proposition 2.12. By Theorem 3.1.7, we obtain in a standard w a y the following result.
: n E N, f l . . . . . Jn E A
C t * ( f ) = s u p { £ i = l II,.(f,)ll
n E N,f~
..... fn-1
~ d N
and 0
=
Jo <~J1 . . . <~/~
= f}.
f]).
Proof. We denote by E ' the topological dual o f E and, i f A C_X, we denote by A~" the weak closure o f A and by c o n v A the convex hull o f A. Let f E ~/. It is sufficient to prove that
Proof. Denote by f i ( f ) the supremum on the rightn hand side. Let J~ . . . . . fn be in ~¢ such that ~ i = l J)
f and set go = 0, gi = ~-~j'=l J) for i/> 1. Then gi C ~ , 0 = g o ~ g l <<. "'" <~g~ = f and gi - g i - I = Ji for i ~< n. Therefore,
cony 2([0, f ] ) C 2([0, f ] ) w .
Let y E c o n v 2 ( [ O , f ] ) and f l . . . . . J ; in E'. Set v = ( f l o 2 . . . . . f , o 2). Then v : , d ~ R n is a nonatomic measure. By Theorem 3.1.7, v([0, f ] ) is convex and
£ II~(Ji)ll = £ II~(gi)- ~(g,-, )11~fi(f), i--1
i--I
from which # * ( f ) ~
fi(f).
A. Avallone, G. Barbieri/Fuzzy Sets and Systems 89 (1997) 231-241
238
N o w let f l . . . . . f , - 1 be in ~ such that 0 = fo ~ f l ~< "'" ~ 1. Then gi E s J and ~i=1 9i = f . Therefore,
£ II~,(J1)- ~(~I-,)11 -- £ II~(g,)ll ~<~,*(f). i=1
i--1
from which f i ( f ) < ~ p * ( f ) .
[]
R e m a r k . Set S = {v : ..4 ---, [0, + o c ] : v is a measure and
II,,(f)ll ~
then
#
is
Proposition 3.2.3. p* • s J --~ [0, + o c ] is a measure.
Proof. Trivially p*(O) = O. Let f , g C d~/ such that f +g
hi= ftA f
and
ki=(ft-
f ) VO.
Then, for every i C {0 . . . . . n}, hi, ki E ~ , + ki, 0 = ho<,hl <~ . . . <~h~ = f and 0 " • • <~kn = g. Therefore,
f i = hi =
Proof. Let fi be like in Section 2. Since [l#(f)l[ <,,u*(f) for every f E d , it is sufficient to prove that, for every e > 0, there exists 6 > 0 such that, if f c ~ ' and f i ( f ) < 3, then p * ( f ) < ~. Let e > 0. Since p* is bounded, by Lemma 3.2.2 we can choose fl . . . . . . fn-1 in ~ ' such that 0 = f0 ~
ko~kl
~,*(1) <
Iip(ft) -/~(J~-,)ll + ~. i=1
Let f ¢
~--~II~(J3) - ,~(f,-,)11
~d. Since # is modular by Proposition 1.9(3),
i=1
n
~
] A ( f i ) -- ~ ( f i - - 1 ) = ~ ( f t " V f )
n
lip(h,) - ,u(h,_,)ll + ~
i=1
-#(ft-I
II,.(k,) - ,u(~,_, )11
i=1
+ #(J~ A f)
V f) - #(J}-I A f),
from which
< P*(f)+P*(9), from which, by Lemma 3.2.2, p * ( f + g ) < ~ # * ( f ) + p*(g).
#*(1) < £
n
Now let e > 0 and f l . . . . . f~, gl . . . . . gm in ~ such t h a t ~-~iL1 ft" = f , ~-~4ml gi = g,
P*(f)
< £
II~'(f, V f ) - ~'(j1-, V f)ll
i-I
+ ~ II,~(f, A f ) - ,.(j1-~ A/)11 +
8
i--]
Set 3 = ~/2n and suppose f i ( f ) < 6. Then ~'~i~1 liP(f/A f ) - P ( J ) - I A f)]l < e/2. Moreover,
I)'(ft)ll + e/2
i=1
and /~*(g)
£ tl~(ft v f ) - ]~(Ji--I V/)11 < £
i--I
II/~(g/)ll + ~/2.
n
i=1
~Z
Then
[P*(ft V f ) - ~*(J)-I V f ) ]
i=1
f f * ( f ) + IJ*(g) < £
II~(J;)ll +
i=1
£
= p*(1) - la*(f). II~(gi)ll + *
i=1
<~p*(f + g) + e.
Therefore/J*(f + g) = p * ( f ) + t~*(g).
Therefore /t*(1) < e + p*(1) - p * ( f ) , from which # * ( f ) < e. N o w we study the range of fuzzy measures with values in B-convex spaces.
A. A v a l l o n e ,
G. B a r b i e r i / F u z z y S e t s a n d S y s t e m s
Definition 3.2.5. E is called B-convex if there exist an integern/>2 and a real number 0 < k < 1 such that, for every x~,... ,x, in E,
~1 . . . . .
~m, w e o b t a i n
min
m i~=l~iXi ~
~,=±1 min :~i=± 1
LO~iXi
239
89 (1997) 231-241
j<~n ~i=-I- 1
t: Z~ixi t=,~/
~
i=1
i<~n
The following is a known example o f B-convex space.
Forj<~n, setAj = {t E N : sj<~t<<,tj}. By the induction assumption, we obtain, for every j <~n, ti
min Example 3.2.6. If E is uniformly convex, then E is B-convex. In fact, we can prove that there exists k E R, with 0 < k < 1, such that, for everyx~,x2 E E,
•i:±1
~-'~ixi
~ n S - l k s-~ supllxill.
~
lEA:
Therefore, m
min
c~i:4-1
~-'~ixi
~
<.kSnS sup Ilxill. i<~m
min II~axl +c~2x2]l ~ 2 k max{llx, ][, [tx2[]}.
:¢,=±1
L e m m a 3.2.8. Suppose that, for every e > 0 and
Trivially, we can suppose max{ ][x~I1, ]Ix211} = 1. Then, if IIx~ - x211~< ½, we obtain
f E d , there exists 9 E ~¢ such that g<~f and lima) - ½~(f)[[ < e. Then I~(.~¢) is convex.
II~,x, +~2x~ll ~< [Ix1 x211~< ½max{llx~ II, tlx211}.
Proof. Let ~,fl E #(sO) and f , g E d such that ~ = /~(f) and fl = / t ( g ) . It is sufficient to prove that
min
:~i=± 1
-
1 Instead, if ]]x~ -x211 > g, by the uniformly convexity o f E we can choose 6 > 0 such that t[(x~ +x2)/2tl < 1 - 6, from which
II~m + ~2xzlt < IIx~ +x211 :~:= 4- I min
~< 2(1-6)max{llx~ [I, IIx211}.
dist (t~( ~¢), ~ - - ~ ) = 0 . Let e > 0. From Lemma 3.1.4, there exist ~b, ~, k in d such that ~ b + O + k ~ < l , f = q ~ + k and 9 = O + k . By assumption, there exist ~b~,~. E ~¢ such that ~b~.~
L e m m a 3.2.7. Suppose E is B-convex and let n and k be as in Definition 3.2.5. l f s E N, m = n s and
11~(4~)-½~(q~)l[ <
X~,...,Xm E E, then
and
8
8
I1~(0~)-½~(0)11 < ~
m
min
~i=±1
i~=l c~ixi ~mkSsup[Ix~ll . i<~m
Set 7 =/~(~b~ + ~ + k). Then 7 E /~(s¢) and
Proof. We use induction. Suppose that the inequality holds for m = n s-I and set m = n s. I f ~ l . . . . . ~m E R, where ~i = +1, and xl,... ,Xm E E, then
m rain
~ti = -~- I
<<- 7
±
i~=l ~iXi ~ min
fl l = -~ l
t!
-
j=l
dist ( # ( sl), ~ - ~ ) # ( f ) 2+ #(g)
fljyj ,
where yj = }-~i=s,, ~ixi,sj = (j - 1)n s-1 + 1 and tj = jn s-l. By the B-convexity o f E and the arbitrarity of
Theorem 3.2.9. Suppose that E is B-convex, ~¢ has
the interpolation property, # is nonatomic and I~* is bounded. Then I~(s¢) is convex.
240
A. Avallone, G. Barbieri/Fuzzy Sets and Systems 89 (1997) 231-241
Proofl Let e > 0. By Lemma 3.2.8, it is sufficient to prove that, for every f E ,~1, there exists g E <~ such that g < ~ f and II#(a)- ½#(f)ll < ~:. Let f E ,~¢. We can suppose # ( f ) ¢ 0. Let k and n like in Definition 3.2.5 and choose s E N such that k S # * ( f ) < e. Set m = n s. By Proposition 3.2.4, Theorem 3.1.3 and Lemma 2.11, we can choose J] . . . . . Jm in .~/ such that fl + " " + Jm = f and #*(Ji) = ( 1 / m ) # * ( f ) for every i<.m. For Qc_ {1 . . . . . m}, set gQ = ~ E Q f~. Then gQ C d and
Let .~ be a clan, G a topological group and # • <~¢ ~ G a measure.
II#(gQ)- ½#(f)ll
Theorem 3.3.2. I f G is complete and # is nonatomic
icQ
= -21
increasing sequence {J;} in ~ , # ( f , + l ) - # ( f , ) ~ 0. p is called locally exhaustive if the restriction of p to every interval of the lattice ~¢ is exhaustive. By Theorem 5.11 o f [15] and Proposition 2.4, the following results hold.
and locally exhaustive, then #(,~1) is connected.
1
=
Definition 3.3.1. # is called exhaustive if, for every
1
7Z#(s;) i~Q
Theorem 3.3.3. I f G is seminormed, g & nonatomic
and ~¢ has the interpolation property, then # ( ~ ) arcwise connected.
icQ
Z#(f')iEQ -- Z#(J})i¢Q
H. Weber has introduced a condition (see * in [15, 2.7],) under which it is possible to characterize the modular functions with relatively weakly compact range. In Lemma 3.3.4, we see that every fuzzy measure on a clan verifies this condition.
"
Therefore,
mjn II#(go)- ½#(s)ll=
is
' <,=±lmin ~,=l ~i/x(f/)
.
L e m m a 3.3.4. Let fo . . . . . f ,
in ~/ such that fo <~
• .. ~ f~. Then, Jor every Then, if we choose g E ~ such that ]l#(g) _ 1 5 # ( f ) l l
I C { 1 ..... n } , ~ - ~ [ # ( J } ) - p ( J )
-- rn~n I I # ( g p ) - 51# ( f ) l l ,
Proof. Let IC_{1 . . . . . n} and, for every i<~n, set gi = J) - ~ - i . Then gi E ,9~ and ~ i E l gi ~< ~ni=l gi = fn - f 0 ~ < 1. Therefore, by Proposition 1.4(1), ~iES gi E ,~ and, by Proposition 1.9(4),
by Lemma 3.2.7 we obtain
N#(g)-~ ~#(f)ll < k'msup
II#(J))ll
i <~m
<.%k S m s u p # * ( f i )
l)]E#(~).
iC1
< ~:.
[]
i <~m
Remark. There exist examples of B-convex spacesvalued measures which verify the assumption o f Theorem 3.2.9, but their range is not convex (see Example 2, p. 262, o f [10] and Example 3.2.6).
[#(Z)
- #(J}-!
)]
iCI
So, by Lemma 3.3.4 and [15, Theorem 4.1], we obtain the following result.
3.3. Other remarks on the range o f f u z z y measures Theorem 3.3.5. Suppose G is a complete locally con-
Since a fuzzy measure is a modular function on a lattice (see Corollary 1.5 and Proposition 1.9(3)), by [15] we obtain other informations on the range.
vex topological linear space. Then # is exhaustive (ff #( d ) is relatively weakly compact. In particular, if G = R, then # is exhaustive iff#(<~¢) is bounded.
A. Avallone, G. Barbieri/Fuzzy Sets and Systems 89 (1997) 231-241
References [1] T. Armstrong and K. Prikry, Liapunoff's theorem for nonatomic finitely additive, bounded, finite-dimensional, vectorvalued measures, Trans. Amer. Math. Soc. 226 (1981) 499 514. [2] Z. Artstein, Set-valued measures, Trans. Amer. Math. Soc. 165 (1972) 103-125. [3] A. Avallone, Liapunov theorem for modular functions, Internat. J. Theoret. Phys. 34 (1995) 1197-1204. [4] A. Avallone and A. Basile, On the Liapunov-Richter theorem in the finitely additive setting, J. Math. Econ. 22 (1993) 557 561. [5] D. Butnariu, Decompositions and range for additive fuzzy measures, Fuzz), Sets and Systems 10 (1983) 135-155. [6] D. Butnariu, Non-atomic fuzzy measures and games, Fuzzy Sets and Systems 17 (1985) 39 52. [7] D. Butnariu and E.P. Klement, Triangular Norm-based Measures and Games with Fuzzy Coalitions (Kluwer, Dordrecht, 1993).
241
[8] D. Candeloro and A. Martellotti, Sul rango di una massa vettoriale, Atti Sem. Mat. Fis. Univ. Modena 28 (1979) 102-111. [9] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summin9 Operators (Cambridge Univ. Press, Cambridge, 1995). [10] J. Diestel and J.J. Uhl, Vector Measures (AMS, Providence, RI, 1979). [11] P. Halmos, The range of a vector measure, Bull. Amer. Math. Soc. 54 (1948) 416-421. [12] V.M. Kadets, Remark on the Liapunov theorem on vector measures, Funct. Anal Appl. 25 (1991) 2 9 5 ~ 9 7 . [13] A. Liapunov, On completely additive vector measures, Izv A k a d Nauk. S S S R 10 (1940) 465-478. [14] H. Volkmer and H. Weber, Der Wertebereich atomloser Inhalte, Arch. Math. 40 (1983) 464 474. [15] H. Weber, On modular functions, Funct. et Approx., to appear.
sets and systems ELSEVIER
Fuzzy Sets and Systems 89 (1997) 231-241
Range of finitely additive fuzzy measures A n n a A v a l l o n e a,*, G i u s e p p i n a B a r b i e r i b a Dipartimento di Matematica, Universita" della Basilicata, Via N. Sauro, 85, 85100 Potenza, Italy b Dipartimento di Matematica, Universita' di Udine, Via delle Scienze, 206, 33100 Udine, Italy
Received August 1995; revised March 1996
Abstract We extend Liapunov theorem to R~-valued finitely additive measures on clans of fuzzy sets. Moreover, we study the convexity o f the range of B-convex-valued measures. (~) 1997 Elsevier Science B.V. Keywords. Measure theory; Fuzzy measures; Range
O. Introduction
In this note, we study some properties of the range of finitely additive fuzzy measures on clans (for a study of fuzzy measures, see for example [7], where finitely additive fuzzy measures are called T~-valuations). Precisely, in the first part, we extend the classical Liapunov theorem concerning the convexity of the range of nonatomic R'-valued measures. This theorem has been proved for a-additive measures on aalgebras in [13] and then in [11]; recently, it has been independently proved for finitely additive measures on algebras with the interpolation property in [8, 14, 1]. Moreover, this result has been proved also for multimeasures on a-algebras (see [2,4]), for modular functions on complemented lattices (see [3]) and for a-additive fuzzy measures on tribes (see [5, 6]). In the second part, we generalize a result of [12] concerning the convexity of the closure of the range of measures with values in B-convex spaces (for a study of these spaces, see for example [9]). * Corresponding author. 0165-0114/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved PH S01 65-011 4( 96 )00099-1
Finally, we conclude with a remark concerning the relative weak compactness of the range of fuzzy measures with values in complete locally convex topological linear spaces.
1. Fuzzy measures
The aim of this section is to give a description of some properties of fuzzy measures. Let X ~ 0 a set. Notation. If f , g E [0, 1] x, we denote by f V g and f A g the functions defined, respectively, by
f V g(x) = m a x { f (x), g(x)}, f A g(x) = min{f(x),g(x)},
x E X,
and by 0 and 1 the functions defined, respectively, by f ( x ) = 0 and f ( x ) = 1 for every x E X. If {f~ : ~ E A} is an increasing (resp. decreasing) net in [0, 1]x and f ( x ) = sup~f~(x) (resp. f ( x ) = inf,(x)) for x E X, then we write f~ T f (resp. f~ ~.f). Moreover, we denote by N and R, respectively, the set of natural numbers and of real numbers.
232
A. Avallone, G. Barbieri/Fuzzy Sets and Systems 89 (1997) 231~41
Definition 1.1. Let .%/~_ [0, 1] x. ,.%/is called clan if (1) 1 c ,%/. (2) I f f , g ~ , % / , t h e n ( f g) VOE~%/. ~%/is called tribe if,%/is a clan and the following property holds: for every sequence { f , } in "%/,(~,eN f , ) A 1 (: ,%/. Proposition 1.2. Let ~%/C [0, 1] x. Then ,%/is a clan if and only if the following conditions hold: (1) I f f C ,%/, then 1 - f ~ ,%/. (2) 0 ~ ,%/. (3) I f f , g ~ ~%/, then ( f + g - 1 ) V 0 ~ ~%/. Proposition 1.3. Let ,%/ be a clan. Then ~%/ is a tribe if and only iJl Jot ever), sequence {f~} in ~%/, [1 - ~,,~l(l - f,)] V 0 ~ ,%/. Proof. We {[1 -- E n Z l ( 1
u s e [~n%l(1 -- f~)] i 1 = - - f n ) ] V O} and Proposition 1.2.
1 --
Definition 1.7. Let ,%/ be a clan and ( G , + ) a semigroup. A function tt : ,%/--~ G is called (fuzzy) measure if the following properties hold: (1) ~(0) = O. (2) I f f , g E ,%/and f + g < ~ l , then # ( f + g) --
~(,1)
+ ~z(g). If G is a topological semigroup, then a measure /~ : ~%/ ---+ G is called a-additive if, for every seoo %/ cx~ quence {f~} E ~%/, with Y'~,,=I ,fn C~ , # ( ~ , = 1 J ~ , ) =
Proposition 1.8. Let ( G, +) be a group, ~%/a clan and IX : ,%/ ---+ G a function such that t~(0) - O. Then # is a measure if and only if, Jor every f , g ~ ,%/,
hi(f+
g - l) v o] + h i ( f +
g) A 1] = n ( f ) + ~ , ( g ) .
Proof. Suppose that # is a measure. Since g = [ ( f ÷ g - - 1) V 0] + [g A (l -- f ) ] a n d ( f + g ) A 1 = f + [g A (1 -- f ) ] , then
Remark. By Propositions 1.2 and 1.3, the definitions of clan and tribe are equivalent, respectively, to the definitions of Too-clan and Too-tribe of [7].
~[(f+
Proposition 1.4. Let ,%/be a clan. Then: n (1) I f f l . . . . . f~ E ,%/, then ( ~ i = l J})A 1 C °~/. (2) ff" f ,g 6 ~%/, then f V g 6 , % / a n d . f l A g 6 ~4.
from which we obtain the desired equality. The converse is trivial. []
Proofl (1) The assertion is true for n = 2, since, by (1.2),
Remark. By Proposition 1.8, the definition of measure, for G = R, is equivalent to the definition of Toovaluation of [7].
g - 1) v o] - ~ ( g ) - ~ [ g A (1 - f ) ] ,
kz[(f + g) A 1] = / ~ ( f )
+ / ~ [ g A (1 -- f ) ] ,
(.fi + .f2)n 1 =l-{[(l-j])+(1-j))-l]VO}E,%/. Then ( 1 ) follows by induction, because
(2) f A g = f - [ ( J - g ) V 0 ] E,%/andfVg= g + [ ( J - g) V 0] C ~ by Proposition 1.2. [] Corollary 1.5. Every clan is a lattice (with respect to the pointwise order).
Proposition 1.9. Let (G, + ) be a group, ¢%/a clan and IZ : , % / ~ G a measure. Then the following properties hold: (1) I f f , g ~ °%/andf<~ g, then I ~ ( g - f ) = P(g)
~(f). (2) f i G = R and ~ 9 0 , then t~ is monotone. (3) # is modular, i. e., Jbr every f , g E ~/, # ( f V g) + ~ ( / A g) = ~(.1) + ~(g). (4) if.t] . . . . . f , C ,%/ and./) + ... + J~,~< 1, then n n ~ ( ~ i : ~ f , ) = E , : , t'(.~). (5) I f G -- R and lt >10, then/~ is subadditive, i.e., /~[(~7-1 ~ ) A 1]~< ~~i~_l #(J}) fi)r every f , . . . . . f~ in.%/.
Proposition 1.6 (Butnariu and Klement [7, Remark 2.8]). Every tribe is a a-complete lattice (with respect to the pointwise order).
ProoL (1) Follows from g = f + (g - f ) . (2) Follows from (1).
233
A. Avallone, G. Barbieri/Fuzzy Sets and Systems 89 (1997) 231-241
(3) S i n c e f = f A g + [ ( f - g ) [ ( f - g) V 0] + g, we obtain p(f) = p(fA
g) + / l [ ( f
V0]andfVg=
-- g) V 0],
gn E d and E k = l Ok = f n ~ 1. Therefore, by Proposition 1.9(4), ~(f) = ~
/ l ( J V g) = p [ ( f - g) V 0] + p(g), from which (3) follows. (4) Follows by induction. (5) By Proposition 1.8, (5) is true for n = 2. Because of (*) in Proposition 1.4, we obtain
Then we use induction.
~(g~) = lim~
[]
(2) ~ (3): Let fn, f E • such that fn ,[ f . Set gn = 1 - f~ and g = 1 - f . Then gn, g E d and g~ T gTherefore, by Proposition 1.9( 1 ), #(1)- #(f)=
# ( g ) = lim/~(gn)---/~(1)- lim #(f~). t/
t/
0 = lim #(gn) = # ( f )
/~-(J') = sup{-#(g) : g E ~,g<~J'} and
- lira ~2~/~(fk)
n k=l
= #(f) - ~
/ ~ + ( f ) = s u p { p ( g ) : g E ,J,g<~ f },
p-(f),J'
= lim~(f~).
k=l
(3) ~ (4): Trivial. (4) ~ (1): Let f , , f E ,~¢ such that f = ~n~_l J~. ii Set g, = f - E k = l fk- Then gn E ~4 and g, I 0. Therefore, by Proposition 1.9(4).
Definition 1.10. Let ,~/be a clan and p : ~ ' --* R a measure. The functions defined by
]#l(f) = P+(f)+
gk
n=l
It(fn) •
[]
n=l
Remark. By Proposition 1.12, the definition of aadditive measure, for G = R, is equivalent to the definition of Too-measure of [7].
E ~/
are called, respectively, positive variation, negative variation and total variation of #. Proposition 1.11 (Butnariu and Klement [7, Propositions 9.3, 9.4 and L e m m a 10.1]). Let d be a clan and p : , ~ --+ R a measure. Then the following properties hold: (1) # + , # - and I#l are measures. (2) # is bounded f f p+ and t~- are bounded iff I/tl is bounded (3) I f p is bounded, then I~ = P+ - # - .
Proposition 1.13. Let ~¢ be a tribe and t~ :su¢ ---+R a a-additive measure, with It >~O. Then # is a-subadditive, i.e. f o r every sequence {fn} in ~1, # [ ( ~ , % 1 f , ) A 1] ~< ~ , % 1 / ~ ( f , ) . Proofl Let {f~} be a sequence in ~/ and set gn = n oo ~ k = l fk and f = ~ n = l fn- Since limn(g~A 1 ) = f A 1, by Propositions 1.9(5) and 1.12, we obtain
~< lim ~-~ # ( f k ) Proposition 1.12. Let ~¢ be a clan, G a topological group and 1~ : ,~ -+ G a measure. Then the following conditions are equivalent: ( 1 ) p is a-additive. (2) f ~ T f , with f n , f E ~ ¢ , ~ / ~ ( f ) : limn #(fn). (3) f , J~f, with f , , f E ~ / , ~ p ( f ) : limn p ( f , ) . (4) f n i O , with f , E ~ ' , ~ lim, # ( f , ) = 0. Proof. (1) ~ (2): Let f . , f E ~ such that fn T f . Set g j = Ji and, for n~>2, gn = f n - f ~ - l . Then
k
1
= ~ #(A).
[]
n=l
2. Nonatomic fuzzy measures In this section, we study some properties of nonatomic fuzzy measures, which will be useful in the next sections.
234
A. Avallone,
G. B a r b i e r i / F u z z y
S e t s a n d S y s t e m s 8 9 ( 1 9 9 7 ) 231 241
Let o~¢ be a clan, (G, + ) a topological group and IX : , ~ ~ G a measure. In the following, i f G is seminormed, we set, for f E ,~,
fi(f)
= sup{Nix(g)ll : g ~
,rd, g4 f } .
Definition 2.1. IX is called n o n a t o m i c (or continuous) if, for every 0-neighbourhood U in G, there exist f l . . . . . f , in ,~¢ such that f l + " " + f ~ = 1 and Ix(g) E U if g E ~¢ and there exists i ~
Definition 2.2. IX is called chained if, for every 0neighbourhood U f ~< g, there exist f 0 ~< f l ~ < ~ < f n 0, tP ~ .~/,0~<~
in G and for every f , g ~ se', f l . . . . . ];_~ in s¢ such that = g and I X ( 6 ) - Ix(q5) ~ and there exists i~
with f = U if that
.D-, ~
Proof. (2) =:> (1): Trivial. (1) ==> (2): Let U be a 0-neighbourhood in G and let x, y E L such that x ~~
r i A x~
1 ~ Si A x i
1 ~ Zi
and 2 is modular, we have 2 ( r ) + 2(zi) = ) . ( Y i ) -~- ).(ri) = 2 ( y i ) + ).(zi-j ) + 2(ri V xi 1) - ),(xi l) and
1
from which 2(s) - 2@) = 2(si V x i - 1 ) - .;t(ri V x i - i ).
Since x i - i ~ ri V x i - 1 ~ , ~ ( s ) - ) . ( r ) E U. []
Si V x i
1 ~ X i , we obtain that
Proposition 2.4. The Jollowing conditions are equivalent: (1) For every f E ~1 a n d f o r every O-neighbourh o o d U in G, there exist ,['1. . . . . fn in ,~¢ such that ,[l + " " + fn = f a n d ix(g) C U i f g E ~¢ and there exists i <~n such that g <~fi. (2) IX is" nonatomic. (3) IX is chained
Proof. (1) ~ (2): Trivial (2) =~ (3): Let U be a 0-neighbourhood in G and choose .fl . . . . . . /~ in <~_¢such that f l + "'" + fn = 1 and Ix(g) ~ U i f g E . ~ and there exists i<~n such that g~
- h ) E U.
r~.
Proposition 2.5. Suppose G s e m i n o r m e d Then: (1) I f f~ . . . . . J'~ E ,~, then ~ [ ( ~ / n 1 f i ) / ~ 1]~< n ~i=1 ~ (fi)" (2) I f IX is" nonatomic, then # is" bounded.
Proof. (1) First suppose n = 2. Let g E o~¢ such that g~<(Ji + J2) A 1 and set h = .fl A g, k = (9 - f l ) V 0. Hence h , k E , ~ ' , h < ~ f l , k < ~ f 2 and g = h + k. Therefore,
2(s) + ;,(z~) = 2(y~) + 2(s,) 2(yi)
A- 2 ( Z i - - I ) -4- 2 ( S i V x i 1 ) - - . ~ ( x i
1),
tl~(g)ll ~< It~(h)ll + II~(k)ll ~<~(fl ) + ~(f2),
A. Avallone, G. Barbieri/ Fuzzy Sets and Systems 89 (1997) 231~41
235
Definition 2.9. Let G = R. I f f E 5u¢, we say that f is a p - atom if p ( f ) # 0 and, i f g E ~¢ and g<~f,
from which fi[(fl + f 2 ) /~ 1] <~fi(f~ ) + fi(f2). Then, using (*) in Proposition 1.4, (1) follows by induction. (2) Let f~ . . . . . f~ be in ,~¢ such that f l + "'" + f~ = 1 and f i ( f l ) < 1 for every i<.n. Then, by (1), for every f ~ ,~¢,[[#(f)[[ ~
then either # ( g ) = 0 or # ( f - g) = 0. W e say that # is atomless if there are not #-atoms. I f G = R n, # is called atomless if each component o f # is atomless.
L e m m a 2.6. Suppose G = R. Then, if f C , ~ ¢ , f i ( f ) ~< l# I ( f ) ~<2 f i ( f ) .
Proposition 2.10. I f G = R, then # is atomless /fflpl & atomless.
Proof. If g E ,~¢ and g<~f, then I # ( g ) t ~ l p l ( f ) , from which we obtain the first inequality. The other inequality follows from # - ( f ) < . . . f i ( f ) and
Proof. W e prove that f c ~¢ is a p - a t o m iff f is a
p+(f)<.fi(f).
[]
Proposition 2.7. Suppose G = R. Then the following conditions are equivalent: ( 1 ) # is nonatomic. (2) Ipl is nonatomic. (3) # + , # are nonatomic. Proof. (1) =~ (2): Let e > 0. By (1), we can choose f l . . . . . fn in .~¢ such that f l + ' " + f~ = 1 and f i ( . ~ ) < ~;/2 for every j-..< n. By L e m m a 2.6, we obtain I P ] ( ~ ) < e for e v e r y j ~ < n . (2) ~ (3): Trivial. (3) =:> (1): Let e > 0 and f l . . . . . fn in ~¢ such that f l + ' " + f ~ = l and p + ( ~ ) < e/2 f o r i < ~ n . By Proposition 2.4, for every j~< n, we can choose Si glj . . . . . g~jj in ,~' such that ~ h = l ghj = J ) and p - ( g h j ) < e/2 for every h <<.sj. Then ~ j = l ~ h s, = l ghj = f and # + ( g h j ) - - . < # + ( ~ ) < e/2. By L e m m a 2.6, we obtain fi(ghj) < ~: for every h<~sj a n d j ~ < n . []
Proposition 2.8. Suppose G = R ~. Then p & nonatomic if and only if each component o f # is nonatomic. Proof. Let p = (#! . . . . . Pn). Suppose that, for every i ~< n, #i is nonatomic. We use induction. Let e > 0. By assumption, p ' = (p 1. . . . . # n- 1) is nonatomic. Then, by L e m m a 2.6, we can choose f l . . . . . fm in ~ such that f l + ' " + J m = 1 and I # i l ( ~ ) < e for i<.n - 1 and j ~
I# I-atom, Suppose f is a [#[-atom. Then we can choose 9 E ,s¢ such the g < ~ f and # ( 9 ) ¢ 0. Hence, by assumption, ] P l ( f - O ) = 0 and therefore # ( g ) = # ( f ) # O. Then f is a p-atom. N o w suppose f is a #-atom. It is sufficient to prove that, i f g E ~¢,g<<.f and # ( g ) = 0, then [pl(g) = 0. For this, we prove that, if h E ~¢ and h ~
Lemma 2.11. Suppose G = R and # >~0. Suppose that, Jor every f E ~¢ and c¢¢ R such that 0 < < p ( f ) , there exists 9 E ~¢ such that g<<.f and #(g) = ~. Then, f o r every f E ~4 a n d m E N, there exist f l . . . . . fm in z¢ such that f l + "'" + fm = f and # ( f i ) = ( 1 / m ) # ( f ) f o r every i <~m. Proof. Let f E ~¢ and m E N, with m>~2. By assumption, we can inductively choose f l . . . . . fro-1 in d such that f l ~ < f , f i ~ < f ~-{fj for i~>2 and # ( f i ) = ( 1 / m ) p ( f ) for every i<.m - 1. Then, m--I m set fm = f - Y ~ i = l fi, fm C d , ~ i = l j ) = f and
#(fro) = ( 1 / m ) p ( f ) .
[]
Proposition 2.12. Suppose G = R". Then the followin 9 conditions hold: (1) I f # is nonatomic, then # is atomless. (2) I f # is a-additive and ~ is a tribe, then p is nonatomic iff # is atomless. Proof. (1) By (2.7), (2.8) and (2.10), we can suppose G = R and p >/0. Let f C ~¢, with p ( f ) :~ 0. Choose e such that 0 < e < p ( f ) and f l . . . . . fm in ~¢ such that f l + " " + f m = f and #(J~) < e for every i ~ m . Since 0 < # ( f ) = ~']~iml#(.1~), there
A. Avallone, G. Barbieri/Fuzzy Sets and Systems 89 (1997) 231~41
236
exists, j~< m such that # ( f j )
-#(j'))
> 0 and # ( f - f j ) >
> 0.
(2) Suppose # is atomless. Like in (1), we can suppose G = R and # >/0. By [6, Theorem 2], the assumption o f L e m m a 2.11 is satisfied. Then, by L e m m a 2.11, # is nonatomic. []
3. Range of fuzzymeasures 3.1. Liapunov theorem In this section, we extend to fuzzy measures the classical Liapunov theorem concerning the convexity o f the range o f nonatomic R"-valued measures. We follow the idea o f [1 1]. Let o~¢ be a clan, n E N and # : ,~¢ --~ R ~ a measure, with # = (#~ . . . . . #~). I f f ~ , ~ ' , we set [0, f ] = {g E ,~¢ : g ~ < / }
Definition 3.1.1. We say that ~ ' has the interpolation property if, for every sequences {f~}, {g~} in ~¢, with f,<~J~+~ <~g,+~ <~g, for every n ~ N, there exists f E ~¢ff such that f~ ~
( 1 ) It is semiconvex. (2) For every f E ~¢ and 2 E [0,1], there exists f;. E zff such that f;<~J', #(f;~) = 2 # ( f ) and f;<~f~, / f 2 --.<2'. (3) For every f E ~¢, # ( [ 0 , f ] ) is" convex. Proof. (1) =~ (2): Let f E ~ ' . By induction, we can obtain, for every dyadic rational k C [0, 1], fk c ~q/ such that fk ~
for every i~< k.
Moreover, by Proposition 3.1.5, for every 2 E [0, 1], we can choose g;~ E ,~¢ such that g;. <~g, ri(g;.) = A'Ci((] ) for every i<~k and g;<~ g;, if2~< 2'. N o w we remark that, if 21 ~<22, then 0 ~ r k + l ( g 2 : ) - zk+l(g~,)<~rk(g& -- g2,) =(22
Lemma 3.1.4. Let f , g , h E ~¢, with g<~ f andh<~ f . Then there exist O , O , k ~ ~ such that 0 + qJ + k <~f ,g = qS + k and h = tp + k. Proof. Set k = g A h , ( h - g ) V 0. []
~b = ( g - h )
V 0 and ~9 =
Proposition 3.1.5. Suppose that ~ has the interpolation property and #i ~ 0 f o r every i <. n. Then the following conditions are equivalent:
(*)
- 21)¢k(g),
from which we obtain that the function 2 E [0, 1] z~+l(g;.) E R is continuous. In a similar way, we obtain the continuity o f the function 2 E [0, 1] ---+ zk+~[(f -- g)l-;~] E R and therefore, set h ;~ = g;. +(f-gh-;., the function ~ ( 2 ) = Zk+l(h ;~) is continuous on [0, 1]. By (*), the assertion is true if rk+l(g) = ½ r k + l ( f ) or r k + l ( f g) = ~1 r * + l ( ] ) - Then we can suppose Zk+l(g) < ½zk+l(f). Since ~(0) = r k + l ( f - g ) , a(1) = zk+i (g) and a is continuous, there
237
A. Avallone, G Barbieri/Fuzzy Sets and Systems 89 (1997) 231~41
exists 20 E [0, 1] such that rk+l(h ;-°) = Moreover, if 1 ~< i ~
l'Ck+l(f).
ri(h ;'° ) = 2ori(g) + (1 - 2o) zi ( f - g)
= lz/(f).
[]
T h e o r e m 3.1.7. S u p p o s e that ~¢ has the interpolation p r o p e r t y and ff is' nonatomic. Then, f o r every f c o~, #([0, f ] ) is convex. Proof. Let f E ~ ' . First suppose ]2i ~0 for every i<<.n and let z be like in Proposition 3.1.6. By Propositions 3.1.5 and 3.1.6, r([0, f ] ) is convex. Therefore, #([0, f ] ) is convex too, because, if we set, for t = (tl . . . . . tn) E R n, T ( t ) = (tl - t2 . . . . . tn--1 -t,, t,), then T is linear and p = T o z. Now we remove the assumption ]Ai ~0 for every i~< n. Set 2 = (Pl+ ,Pl , ' " , / ~ n+, / 4 - ) " By Propositions 2.7 and 2.8, 2 is nonatomic. Then ) ~ ( [ 0 , f ] ) is convex. Hence, p([0, f ] ) is convex too, because, if we set, for t = ( h . . . . . t2n) E R 2~, T ( t ) = (tl - t2,t2 - t3 . . . . . t2n-l - t2~), then T is linear and IL =
To)..
therefore ( f l ( Y ) . . . . . f . ( y ) ) E v([0, f ] ) . Then there exists g E o~¢ such that g < . f and J } ( y ) = f i ( 2 ( g ) ) f o r e v e r y i ~ n . Hence y E 2([0, f ] ) w. [] 3.2. R a n g e o f f u z z y measures with values in B - c o n v e x spaces
We shall extend to fuzzy measures a result o f [12] concerning the range o f measures with values in Bconvex spaces. Let E be a normed linear space, d a clan and p : ~¢ ~ E a measure. First we prove that there exists a measure #* : .~---~ [0, + o c ] such that, for every f E , d , [[P(f)l[ ~< Y ( f ) . This is a tool in the p r o o f o f Theorem 3.2.9. Definition 3.2.1. If f E s~', we set
n
}
z
"=
R e m a r k . By [7, Proposition 15.10], if E = R, p* is the total variation o f p. L e m m a 3.2.2. If f E ~4, then if(f)
= sup { £" ,l~(Ji) / = - 1~(J~-,)l
C o r o l l a r y 3.1.8. S u p p o s e that A has the interpolation property, E is a locally c o n v e x H a u s d o r f f topological linear space a n d ), : ~4 -+ E is" a n o n a t o m i c measure. Then, J o t ever)' f E ~ , the w e a k closure o f ) , ( [ 0 , f ] ) is" equal to the closed convex hull
oj;.([o,
~f~ f
and
[]
R e m a r k . Theorem 3.1.7 is a generalization o f Theorem 1 o f [6] about convexity, because every tribe has the interpolation property (since it is a acomplete lattice by Proposition 1.6) and because o f Proposition 2.12. By Theorem 3.1.7, we obtain in a standard w a y the following result.
: n E N, f l . . . . . Jn E A
C t * ( f ) = s u p { £ i = l II,.(f,)ll
n E N,f~
..... fn-1
~ d N
and 0
=
Jo <~J1 . . . <~/~
= f}.
f]).
Proof. We denote by E ' the topological dual o f E and, i f A C_X, we denote by A~" the weak closure o f A and by c o n v A the convex hull o f A. Let f E ~/. It is sufficient to prove that
Proof. Denote by f i ( f ) the supremum on the rightn hand side. Let J~ . . . . . fn be in ~¢ such that ~ i = l J)
f and set go = 0, gi = ~-~j'=l J) for i/> 1. Then gi C ~ , 0 = g o ~ g l <<. "'" <~g~ = f and gi - g i - I = Ji for i ~< n. Therefore,
cony 2([0, f ] ) C 2([0, f ] ) w .
Let y E c o n v 2 ( [ O , f ] ) and f l . . . . . J ; in E'. Set v = ( f l o 2 . . . . . f , o 2). Then v : , d ~ R n is a nonatomic measure. By Theorem 3.1.7, v([0, f ] ) is convex and
£ II~(Ji)ll = £ II~(gi)- ~(g,-, )11~fi(f), i--1
i--I
from which # * ( f ) ~
fi(f).
A. Avallone, G. Barbieri/Fuzzy Sets and Systems 89 (1997) 231-241
238
N o w let f l . . . . . f , - 1 be in ~ such that 0 = fo ~ f l ~< "'" ~
£ II~,(J1)- ~(~I-,)11 -- £ II~(g,)ll ~<~,*(f). i=1
i--1
from which f i ( f ) < ~ p * ( f ) .
[]
R e m a r k . Set S = {v : ..4 ---, [0, + o c ] : v is a measure and
II,,(f)ll ~
then
#
is
Proposition 3.2.3. p* • s J --~ [0, + o c ] is a measure.
Proof. Trivially p*(O) = O. Let f , g C d~/ such that f +g
hi= ftA f
and
ki=(ft-
f ) VO.
Then, for every i C {0 . . . . . n}, hi, ki E ~ , + ki, 0 = ho<,hl <~ . . . <~h~ = f and 0 " • • <~kn = g. Therefore,
f i = hi =
Proof. Let fi be like in Section 2. Since [l#(f)l[ <,,u*(f) for every f E d , it is sufficient to prove that, for every e > 0, there exists 6 > 0 such that, if f c ~ ' and f i ( f ) < 3, then p * ( f ) < ~. Let e > 0. Since p* is bounded, by Lemma 3.2.2 we can choose fl . . . . . . fn-1 in ~ ' such that 0 = f0 ~
ko~kl
~,*(1) <
Iip(ft) -/~(J~-,)ll + ~. i=1
Let f ¢
~--~II~(J3) - ,~(f,-,)11
~d. Since # is modular by Proposition 1.9(3),
i=1
n
~
] A ( f i ) -- ~ ( f i - - 1 ) = ~ ( f t " V f )
n
lip(h,) - ,u(h,_,)ll + ~
i=1
-#(ft-I
II,.(k,) - ,u(~,_, )11
i=1
+ #(J~ A f)
V f) - #(J}-I A f),
from which
< P*(f)+P*(9), from which, by Lemma 3.2.2, p * ( f + g ) < ~ # * ( f ) + p*(g).
#*(1) < £
n
Now let e > 0 and f l . . . . . f~, gl . . . . . gm in ~ such t h a t ~-~iL1 ft" = f , ~-~4ml gi = g,
P*(f)
< £
II~'(f, V f ) - ~'(j1-, V f)ll
i-I
+ ~ II,~(f, A f ) - ,.(j1-~ A/)11 +
8
i--]
Set 3 = ~/2n and suppose f i ( f ) < 6. Then ~'~i~1 liP(f/A f ) - P ( J ) - I A f)]l < e/2. Moreover,
I)'(ft)ll + e/2
i=1
and /~*(g)
£ tl~(ft v f ) - ]~(Ji--I V/)11 < £
i--I
II/~(g/)ll + ~/2.
n
i=1
~Z
Then
[P*(ft V f ) - ~*(J)-I V f ) ]
i=1
f f * ( f ) + IJ*(g) < £
II~(J;)ll +
i=1
£
= p*(1) - la*(f). II~(gi)ll + *
i=1
<~p*(f + g) + e.
Therefore/J*(f + g) = p * ( f ) + t~*(g).
Therefore /t*(1) < e + p*(1) - p * ( f ) , from which # * ( f ) < e. N o w we study the range of fuzzy measures with values in B-convex spaces.
A. A v a l l o n e ,
G. B a r b i e r i / F u z z y S e t s a n d S y s t e m s
Definition 3.2.5. E is called B-convex if there exist an integern/>2 and a real number 0 < k < 1 such that, for every x~,... ,x, in E,
~1 . . . . .
~m, w e o b t a i n
min
m i~=l~iXi ~
~,=±1 min :~i=± 1
LO~iXi
239
89 (1997) 231-241
j<~n ~i=-I- 1
t: Z~ixi t=,~/
~
i=1
i<~n
The following is a known example o f B-convex space.
Forj<~n, setAj = {t E N : sj<~t<<,tj}. By the induction assumption, we obtain, for every j <~n, ti
min Example 3.2.6. If E is uniformly convex, then E is B-convex. In fact, we can prove that there exists k E R, with 0 < k < 1, such that, for everyx~,x2 E E,
•i:±1
~-'~ixi
~ n S - l k s-~ supllxill.
~
lEA:
Therefore, m
min
c~i:4-1
~-'~ixi
~
<.kSnS sup Ilxill. i<~m
min II~axl +c~2x2]l ~ 2 k max{llx, ][, [tx2[]}.
:¢,=±1
L e m m a 3.2.8. Suppose that, for every e > 0 and
Trivially, we can suppose max{ ][x~I1, ]Ix211} = 1. Then, if IIx~ - x211~< ½, we obtain
f E d , there exists 9 E ~¢ such that g<~f and lima) - ½~(f)[[ < e. Then I~(.~¢) is convex.
II~,x, +~2x~ll ~< [Ix1 x211~< ½max{llx~ II, tlx211}.
Proof. Let ~,fl E #(sO) and f , g E d such that ~ = /~(f) and fl = / t ( g ) . It is sufficient to prove that
min
:~i=± 1
-
1 Instead, if ]]x~ -x211 > g, by the uniformly convexity o f E we can choose 6 > 0 such that t[(x~ +x2)/2tl < 1 - 6, from which
II~m + ~2xzlt < IIx~ +x211 :~:= 4- I min
~< 2(1-6)max{llx~ [I, IIx211}.
dist (t~( ~¢), ~ - - ~ ) = 0 . Let e > 0. From Lemma 3.1.4, there exist ~b, ~, k in d such that ~ b + O + k ~ < l , f = q ~ + k and 9 = O + k . By assumption, there exist ~b~,~. E ~¢ such that ~b~.~
L e m m a 3.2.7. Suppose E is B-convex and let n and k be as in Definition 3.2.5. l f s E N, m = n s and
11~(4~)-½~(q~)l[ <
X~,...,Xm E E, then
and
8
8
I1~(0~)-½~(0)11 < ~
m
min
~i=±1
i~=l c~ixi ~mkSsup[Ix~ll . i<~m
Set 7 =/~(~b~ + ~ + k). Then 7 E /~(s¢) and
Proof. We use induction. Suppose that the inequality holds for m = n s-I and set m = n s. I f ~ l . . . . . ~m E R, where ~i = +1, and xl,... ,Xm E E, then
m rain
~ti = -~- I
<<- 7
±
i~=l ~iXi ~ min
fl l = -~ l
t!
-
j=l
dist ( # ( sl), ~ - ~ ) # ( f ) 2+ #(g)
fljyj ,
where yj = }-~i=s,, ~ixi,sj = (j - 1)n s-1 + 1 and tj = jn s-l. By the B-convexity o f E and the arbitrarity of
Theorem 3.2.9. Suppose that E is B-convex, ~¢ has
the interpolation property, # is nonatomic and I~* is bounded. Then I~(s¢) is convex.
240
A. Avallone, G. Barbieri/Fuzzy Sets and Systems 89 (1997) 231-241
Proofl Let e > 0. By Lemma 3.2.8, it is sufficient to prove that, for every f E ,~1, there exists g E <~ such that g < ~ f and II#(a)- ½#(f)ll < ~:. Let f E ,~¢. We can suppose # ( f ) ¢ 0. Let k and n like in Definition 3.2.5 and choose s E N such that k S # * ( f ) < e. Set m = n s. By Proposition 3.2.4, Theorem 3.1.3 and Lemma 2.11, we can choose J] . . . . . Jm in .~/ such that fl + " " + Jm = f and #*(Ji) = ( 1 / m ) # * ( f ) for every i<.m. For Qc_ {1 . . . . . m}, set gQ = ~ E Q f~. Then gQ C d and
Let .~ be a clan, G a topological group and # • <~¢ ~ G a measure.
II#(gQ)- ½#(f)ll
Theorem 3.3.2. I f G is complete and # is nonatomic
icQ
= -21
increasing sequence {J;} in ~ , # ( f , + l ) - # ( f , ) ~ 0. p is called locally exhaustive if the restriction of p to every interval of the lattice ~¢ is exhaustive. By Theorem 5.11 o f [15] and Proposition 2.4, the following results hold.
and locally exhaustive, then #(,~1) is connected.
1
=
Definition 3.3.1. # is called exhaustive if, for every
1
7Z#(s;) i~Q
Theorem 3.3.3. I f G is seminormed, g & nonatomic
and ~¢ has the interpolation property, then # ( ~ ) arcwise connected.
icQ
Z#(f')iEQ -- Z#(J})i¢Q
H. Weber has introduced a condition (see * in [15, 2.7],) under which it is possible to characterize the modular functions with relatively weakly compact range. In Lemma 3.3.4, we see that every fuzzy measure on a clan verifies this condition.
"
Therefore,
mjn II#(go)- ½#(s)ll=
is
' <,=±lmin ~,=l ~i/x(f/)
.
L e m m a 3.3.4. Let fo . . . . . f ,
in ~/ such that fo <~
• .. ~ f~. Then, Jor every Then, if we choose g E ~ such that ]l#(g) _ 1 5 # ( f ) l l
I C { 1 ..... n } , ~ - ~ [ # ( J } ) - p ( J )
-- rn~n I I # ( g p ) - 51# ( f ) l l ,
Proof. Let IC_{1 . . . . . n} and, for every i<~n, set gi = J) - ~ - i . Then gi E ,9~ and ~ i E l gi ~< ~ni=l gi = fn - f 0 ~ < 1. Therefore, by Proposition 1.4(1), ~iES gi E ,~ and, by Proposition 1.9(4),
by Lemma 3.2.7 we obtain
N#(g)-~ ~#(f)ll < k'msup
II#(J))ll
i <~m
<.%k S m s u p # * ( f i )
l)]E#(~).
iC1
< ~:.
[]
i <~m
Remark. There exist examples of B-convex spacesvalued measures which verify the assumption o f Theorem 3.2.9, but their range is not convex (see Example 2, p. 262, o f [10] and Example 3.2.6).
[#(Z)
- #(J}-!
)]
iCI
So, by Lemma 3.3.4 and [15, Theorem 4.1], we obtain the following result.
3.3. Other remarks on the range o f f u z z y measures Theorem 3.3.5. Suppose G is a complete locally con-
Since a fuzzy measure is a modular function on a lattice (see Corollary 1.5 and Proposition 1.9(3)), by [15] we obtain other informations on the range.
vex topological linear space. Then # is exhaustive (ff #( d ) is relatively weakly compact. In particular, if G = R, then # is exhaustive iff#(<~¢) is bounded.
A. Avallone, G. Barbieri/Fuzzy Sets and Systems 89 (1997) 231-241
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