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On the Dobrakov submeasure on fuzzy sets
Fuzzy Sets and Systems 151 (2005) 635 – 641 www.elsevier.com/locate/fss
On the Dobrakov submeasure on fuzzy sets Beloslav Rieˇcan Slovak Academy of Sciences, Mathematical Institute, Stefanikova 49, 81473 Bratislava, Slovakia Received 29 January 2003; received in revised form 29 June 2004; accepted 26 July 2004 Available online 13 August 2004
Abstract The well-known fact that any nonatomic measure has the Darboux property has been generalized in many directions. Particularly by Dobrakov (Dissertationes Math. 112 (1974) 1) and recently by Klimkin and Svistula (Mat. Sb. 192 (2001) 41). In this paper the fuzzy sets are taken instead of sets as elements of domain of considered mappings. © 2004 Elsevier B.V. All rights reserved. Keywords: Fuzzy sets; Submeasures; Darboux property; Nonatomicity
1. Introduction A measure : S → [0, ∞) defined on a -algebra S is called nonatomic if for every E ∈ S with 0 < (E) there exists F ∈ S , F ⊂ E such that 0 < (F ) < (E). If is nonatomic, then it has the Darboux property, i.e. to any E ∈ S and any t ∈ R satisfying 0 < t < (E) there exists F ∈ S , F ⊂ E such that (F ) = t (see [5,8]). The result has been generalized in many directions ([13,14,16,17]), one of the most original setting was suggested by Dobrakov ([6,7]; see [12] for recent development). Recall that in spaces endowed with a Dobrakov submeasure, many deep results of the classical measure theory hold, e.g. Jegorov’s theorem, Lusin’s theorem, Riesz–Fischer’ s theorem, etc. (see [3,4,6,7,19]). In the paper we shall work with fuzzy sets, i.e. with functions f : X → [0, 1] (see [1,2,10,18]). The study of submeasures has brought nontrivial generalizations of the theory of -additive measures. In the paper we show that also the fuzzy version of submeasures can lead to some profound results with expected applications. E-mail address: [email protected] (B. Rieˇcan). 0165-0114/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2004.07.004
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2. Formulation We shall define the fuzzy variant of the Dobrakov submeasure in a very general form. Therefore first we give some axioms on the domain of (Definition 1) and then some axioms on . In both cases some examples will be presented. Definition 2.1. We shall say that a family F ⊂ [0, ∞)X is an F-domain, if it satisfies the following conditions: 1. if f, g ∈ F , then f ∧ g ∈ F . 2. if f, g ∈ F , g f , then f − g ∈ F . ∞ 3. if fn ∈ F (n = 1, 2, . . .), f ∈ F and ∞ n=1 fn f , then n=1 fn ∈ F . 4. if gn ∈ F (n = 1, 2, . . .), and gn g ∈ [0, ∞), then g ∈ F . Examples 1. The set F of all characteristic functions {A ; A ∈ S }, where S is a -algebra of subsets of X. 2. The family of all fuzzy subsets of X, i.e. F = [0, 1]X . 3. The set F of all non-negative measurable functions on a measure space (X, S ). 4. The set F of all non-negative integrable functions defined on a probability space (X, S , P ). 5. The set F of all measurable fuzzy events with respect to some -algebra. More general any T-tribe with respect to strict Frank Ts -norm (see [15]) or anyTL -tribe (see [11]).
Definition 2.2. A Dobrakov submeasure is a function : F → [0, ∞) defined on an F-domain F and satisfying the following conditions: (i) if f, g ∈ F , f g, then (f ) (g); (ii) to any f ∈ F and any ε > 0 there exists > 0 such that f + g ∈ F and (f + g) (f ) + ε, whenever g ∈ F , (g) < ; (iii) if fn ∈ F , fn 0, then (fn ) 0. Examples 1. If m is a Dobrakov submeasure on a -algebra S (in the sense of [12]), F = {A ; A ∈ S }, (A ) = m(A), then is a Dobrakov submeasure in the sense of Definition 1. Recall that property (ii) is a very useful generalization of usual subadditivity (A ∪ B) (A) + (B). Similar but different is the concept of autocontinuous functions [18]; of course the domain of the functions are not fuzzy sets, but only sets. 2. Evidently additivity implies subadditivity, hence any measure induces a submeasure. Moreover, also some T-measures [1,2] present Dobrakov submeasures, e.g. TL -measures with the Lukasiewicz norm TL . Recall that : F → [0, 1] is TL -measure on a TL -tribe F , if (1 ) = 1, (iii) of Definition 2 holds, and (f ) = (g) + (h), whenever f, g, h ∈ F , and f = g + h. It is interesting that [2] contains also an equivalence theorem about Darboux property and nonatomicity, of course only in the additive case. On the contrary, we prove the equivalence also in subadditive case. 3. If m is a subadditive measure, then the integral (f ) = J0 (f ) with respect to m [19, Theorem 4.3.2] is a subadditive measure, hence a Dobrakov submeasure, too.
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4. Square of any Dobrakov submeasure m (e.g. a finite additive measure) is a Dobrakov submeasure. Indeed, m2 (f + g) (m(f ) + ε)2 = m2 (f ) + 2m(f )ε + ε 2 3ε2 for sufficiently small ε.
Definition 2.3. A Dobrakov submeasure is nonatomic, if to any f ∈ F with (f ) > 0 there exists g ∈ F such that g f and 0 < (g) < (f ). Definition 2.4. A Dobrakov submeasure has the Saks property (see [20]), if for any ε > 0 and any f ∈ F there exists an ε- partition of f, i.e. such elements f1 , f2 , . . . , fk ∈ F that ki=1 fi = f , and (fi ) < ε (i = 1, 2, . . . , k). Definition 2.5. A Dobrakov submeasure has the Darboux property, if to any f ∈ F with (f ) > 0 and any t ∈ (0, (f )) there exists g ∈ F such that g f and (g) = t.
3. Main result Theorem 3.1. For any Dobrakov submeasure the following three properties are equivalent: (i) is nonatomic; (ii) has the Saks property; (iii) has the Darboux property. Since the Darboux property evidently implies nonatomicity, the proof of theorem follows from the following two propositions. The main ideas of the proof are taken from [11]. Lemma 3.1. If a Dobrakov submeasure is nonatomic, and f ∈ F , (f ) > 0, then to any ε > 0 there exists h ∈ F such that h f and 0 < (h) < ε. Proof. Contrary, assume that there exists ε > 0 such that for any h ∈ F , h f either (h) ε or (h) = 0. Since is nonatomic, there exists h1 ∈ F , h1 f such that 0 < (h1 ) < (f ). By preceding (h1 ) ε. Consider f − h1 . Evidently 0 f − h1 f . By assumption 2 of Definition 2.1, f − h1 ∈ F . We want to prove that (f − h1 ) > 0. Put = 1/2((f ) − (h1 )) > 0. By (ii) of Definition 2.3 there exists > 0 such that (h1 + k) < (h1 ) +
whenever k ∈ F , (k) < . If (f − h1 ) = 0, then (put k = f − h1 ) (f ) = (h1 + f − h1 ) (h1 ) + = (h1 ) + 21 ((f ) − (h1 ))
= 21 ((f ) + (h1 )) < (f ), which is a contradiction. Hence (f − h1 ) > 0.
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We have proved that there exists h2 f − h we obtain a 1 with (h2 ) ε. Continuing the process ∞ ∞ sequence (hn )∞ of elements of F such that h f and (h ) ε. Put g = n n n n=1 i=n hi . Then n=1 gn ∈ F (n = 1, 2, . . .), gn 0. Since then (gn ) 0, there exists such n that (hn ) (gn ) < ε,
which is a contradiction.
Proposition 3.1. If a Dobrakov submeasure is nonatomic, then has the Saks property. Proof. Let g ∈ F , ε > 0. We want to construct an ε-partition of g. Put a1 = sup{(f ); f ∈ F , f g, (f ) < ε}. If a1 = 0, then (g) = 0 < ε and the proof is finished. If a1 > 0, then there exists g1 ∈ F , g1 g such that a1 < (g1 ) a1 . (g1 ) < ε, 2 If (g − g1 ) < ε the proof is complete. If not, put a2 = sup{(f ); f ∈ F , f g − g1 , (f ) < ε} and construct g2 ∈ F , g2 g − g1 such that (g2 ) < ε,
a2 < (g2 ) a2 . 2
By such a way a sequence (gn ) (possibly infinite) can be constructed such that an < (gn ) an = sup{(f ); f ∈ F , f g − g1 − · · · − gn−1 , (f ) < ε}. 2 Put hn = ∞ i=n gi . Then 0<
0<
an < (gn ) (hn ). 2
Since hn g, hn 0, we obtain (hn ) 0 and hence (1) limn→∞ a n = 0. Put f = g − ∞ i=1 gi . We want to prove that (f ) = 0. On the contrary let (f ) > 0. By Lemma 3.1 there exists h ∈ F , h f such that 0 < (h) < ε. Of course, hf = g −
∞ i=1
gi g −
k
gi (k = 1, 2, . . .).
i=1
Since, moreover, (h) < ε, we obtain (2) (h) ak (k = 1, 2, . . .). By (1) and (2) we conclude that (h) 0, which is a contradiction.
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Since ∞ i=n gi 0 there exists n0 such that ∞ gi < ε. i=n0
Consider the family ∞ A = g1 , g2 , . . . , gn0 −1 , gi , f . i=n0
Then (k) < ε for any k ∈ F , and moreover
∞ ∞ g1 + · · · + gn0 −1 + gi + g − gi = g, i=n0
hence A is an ε-partition of g.
i=1
Proposition 3.2. If a Dobrakov submeasure has the Saks property, then has also the Darboux property. Proof. Let f ∈ F , 0 < t < (f ). Let (εn ) be a sequence of real numbers, εn 0, εn < t. Because of the Saks property there exists an ε1 -partition {f1 , . . . , fk } of f. Since (f1 ) < ε1 < t, but (f1 + · · · + fk ) = (f ) > t, there exists l such that (f1 + · · · + fl ) < t, (f1 + · · · + fl + fl+1 ) t.
Put g1 = f1 + · · · + fl , h1 = f1 + · · · + fl + fl+1 . Then g1 h1 f, (g1 ) < t, (h1 ) t, (h1 − g1 ) = (fl+1 ) < ε1 .
Apply now the Saks property to the function h1 − g1 . There exist k1 , . . . , kn ∈ F such that h1 − g1 =
n
ki , (ki ) < ε2
(i = 1, . . . , n).
i=1
Again, since (g1 ) < t, (g1 + k1 + · · · + kn ) = (g1 + (h1 − g1 )) t, there exists u such that (g1 + k1 + · · · + ku ) < t, (g1 + k1 + · · · + ku + ku+1 ) t.
Put g2 = g1 + k1 + · · · + ku , h2 = g1 + k1 + · · · + ku + ku+1 . Then g1 g2 h2 h1 f, (g2 ) < t, (h2 ) t, (h2 − g2 ) = (ku+1 ) < ε2 .
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By this method we can obtain two sequences (gn ), (hn ) of elements of F such that g1 g2 · · · gn hn · · · h2 h1 f, (hn − gn ) < εn
(n = 1, 2, . . .).
Put g=
∞
gn .
n=1
Then g ∈ F , g f . We want to prove that (g) = t. First we shall show that the inequality (g) < t is impossible. If (g) < t, then we can put ε = t − (g) > 0. By (ii) there exists > 0 such that k + g ∈ F , (g + k) (g) + ε, whenever k ∈ F and (k) < . Since εn 0, there exists n0 such that (hn0 − gn0 ) εn0 < .
Then (hn0 ) = (gn0 + (hn0 − gn0 )) (g + (hn0 − gn0 )) < (g) + ε = t,
which is a contradiction. We have proved that (g) t. Let ε > 0. By (ii) there exists > 0 such that for any k ∈ F with (k) < we obtain that gn0 + k ∈ F and (gn0 + k) (gn0 ) + ε. Since (g − gn0 ) (hn0 − gn0 ) < εn0 < ,
we have (g) = (gn0 + (g − gn0 )) < (gn0 ) + ε < t + ε.
Since (g) < t + ε for any ε > 0, we have (g) t, and therefore (g) = t.
4. Concluding remarks The results present contributions in two directions. Firstly, they are new and in some sense surprising from the point of view of fuzzy sets theory: the fact that the range of a given mapping is a convex set does not depend on the additivity of the given mapping, but only on a very weak kind of subadditivity. Secondly, from the mathematical point of view one can recognize new methods of proofs based on the Dobrakov ideas. For example they can be rearranged for the MV-algebra domains [9]. Acknowledgements Supported by grant VEGA 1/9056/02.
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References [1] D. Butnariu, E.P. Klement, Triangular Norm-Based Measures and Games with Fuzzy Coalitions, Kluwer, Dordrecht, 1993. [2] D. Butnariu, E.P. Klement, Triangular, norm-based measures, in: E. Pap (Ed.), Handbook on Measure Theory, Elsevier, Amsterdam, 2002, pp. 947–1010. ˇ [3] P. Cernek, Product of submeasures, Acta Math. Univ. Comenian 40–41 (1982) 301–308. ˇ [4] P. Cernek, The least bound of the additive measures and integrals, Zb. Rad. Prirod. Mat. Fak. Ser. Mat. 24 (1) (1994) 73–80. [5] N. Dinculeanu, Vector Measures, VEB, Berlin, 1966. [6] I. Dobrakov, On submeasures I, Dissertationes Math. 112 (1974) 1–35. [7] I. Dobrakov, J. Dobrakovová, On submeasures II, Math. Slovaca 30 (1980) 65–81. [8] P.R. Halmos, The range of a vector measure, Bull. Amer. Math. Soc. 54 (1948) 416–421. [9] M. Jureˇcková, F. Chovanec, On some properties of submeasures on MV-algebras, Math. Slovaca 54 (2004) 161–167. [10] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer, Dordrecht, 2000. [11] E.P. Klement, M. Navara, A characterization of tribes with respect to the Lukasiewicz t-norm, Czechoslovak Math. J. 47 (1997) 689–700. [12] V.M. Klimkin, M.G. Svistula, The Darboux property of non-additive set functions, Mat. Sb. 192 (2001) 41–50 (in Russian). [13] D. Maharam, Finitely additive measures on integers, Sankya Ser. A 38 (1976) 44–49. [14] A. Martelotti, Topological properties of the range of a group-valued finitely additive measure, J. Math. Anal. Appl. 110 (1985) 411–424. [15] R. Mesiar, M. Navara, Ts -tribes and Ts -measures, J. Math. Anal. Appl. 201 (1996) 91–102. [16] K. Musial, Absolute continuity and the range of group valued measure, Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys. 21 (1973) 105–113. [17] V. Olejˇcek, Darboux property of finitely additive measure on -ring, Math. Slovaca 27 (1977) 195–201. [18] E. Pap, Null-Additive Set Functions, Kluwer, Dordrecht, 1995. [19] B. Rieˇcan, T. Neubrunn, Integral, Measure, and Ordering, Kluwer, Dordrecht, 1997. [20] S. Saks, Addition to the note on some functionals, Trans. Amer. Math. Soc. 35 (1933) 967–974.
On the Dobrakov submeasure on fuzzy sets Beloslav Rieˇcan Slovak Academy of Sciences, Mathematical Institute, Stefanikova 49, 81473 Bratislava, Slovakia Received 29 January 2003; received in revised form 29 June 2004; accepted 26 July 2004 Available online 13 August 2004
Abstract The well-known fact that any nonatomic measure has the Darboux property has been generalized in many directions. Particularly by Dobrakov (Dissertationes Math. 112 (1974) 1) and recently by Klimkin and Svistula (Mat. Sb. 192 (2001) 41). In this paper the fuzzy sets are taken instead of sets as elements of domain of considered mappings. © 2004 Elsevier B.V. All rights reserved. Keywords: Fuzzy sets; Submeasures; Darboux property; Nonatomicity
1. Introduction A measure : S → [0, ∞) defined on a -algebra S is called nonatomic if for every E ∈ S with 0 < (E) there exists F ∈ S , F ⊂ E such that 0 < (F ) < (E). If is nonatomic, then it has the Darboux property, i.e. to any E ∈ S and any t ∈ R satisfying 0 < t < (E) there exists F ∈ S , F ⊂ E such that (F ) = t (see [5,8]). The result has been generalized in many directions ([13,14,16,17]), one of the most original setting was suggested by Dobrakov ([6,7]; see [12] for recent development). Recall that in spaces endowed with a Dobrakov submeasure, many deep results of the classical measure theory hold, e.g. Jegorov’s theorem, Lusin’s theorem, Riesz–Fischer’ s theorem, etc. (see [3,4,6,7,19]). In the paper we shall work with fuzzy sets, i.e. with functions f : X → [0, 1] (see [1,2,10,18]). The study of submeasures has brought nontrivial generalizations of the theory of -additive measures. In the paper we show that also the fuzzy version of submeasures can lead to some profound results with expected applications. E-mail address: [email protected] (B. Rieˇcan). 0165-0114/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2004.07.004
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2. Formulation We shall define the fuzzy variant of the Dobrakov submeasure in a very general form. Therefore first we give some axioms on the domain of (Definition 1) and then some axioms on . In both cases some examples will be presented. Definition 2.1. We shall say that a family F ⊂ [0, ∞)X is an F-domain, if it satisfies the following conditions: 1. if f, g ∈ F , then f ∧ g ∈ F . 2. if f, g ∈ F , g f , then f − g ∈ F . ∞ 3. if fn ∈ F (n = 1, 2, . . .), f ∈ F and ∞ n=1 fn f , then n=1 fn ∈ F . 4. if gn ∈ F (n = 1, 2, . . .), and gn g ∈ [0, ∞), then g ∈ F . Examples 1. The set F of all characteristic functions {A ; A ∈ S }, where S is a -algebra of subsets of X. 2. The family of all fuzzy subsets of X, i.e. F = [0, 1]X . 3. The set F of all non-negative measurable functions on a measure space (X, S ). 4. The set F of all non-negative integrable functions defined on a probability space (X, S , P ). 5. The set F of all measurable fuzzy events with respect to some -algebra. More general any T-tribe with respect to strict Frank Ts -norm (see [15]) or anyTL -tribe (see [11]).
Definition 2.2. A Dobrakov submeasure is a function : F → [0, ∞) defined on an F-domain F and satisfying the following conditions: (i) if f, g ∈ F , f g, then (f ) (g); (ii) to any f ∈ F and any ε > 0 there exists > 0 such that f + g ∈ F and (f + g) (f ) + ε, whenever g ∈ F , (g) < ; (iii) if fn ∈ F , fn 0, then (fn ) 0. Examples 1. If m is a Dobrakov submeasure on a -algebra S (in the sense of [12]), F = {A ; A ∈ S }, (A ) = m(A), then is a Dobrakov submeasure in the sense of Definition 1. Recall that property (ii) is a very useful generalization of usual subadditivity (A ∪ B) (A) + (B). Similar but different is the concept of autocontinuous functions [18]; of course the domain of the functions are not fuzzy sets, but only sets. 2. Evidently additivity implies subadditivity, hence any measure induces a submeasure. Moreover, also some T-measures [1,2] present Dobrakov submeasures, e.g. TL -measures with the Lukasiewicz norm TL . Recall that : F → [0, 1] is TL -measure on a TL -tribe F , if (1 ) = 1, (iii) of Definition 2 holds, and (f ) = (g) + (h), whenever f, g, h ∈ F , and f = g + h. It is interesting that [2] contains also an equivalence theorem about Darboux property and nonatomicity, of course only in the additive case. On the contrary, we prove the equivalence also in subadditive case. 3. If m is a subadditive measure, then the integral (f ) = J0 (f ) with respect to m [19, Theorem 4.3.2] is a subadditive measure, hence a Dobrakov submeasure, too.
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4. Square of any Dobrakov submeasure m (e.g. a finite additive measure) is a Dobrakov submeasure. Indeed, m2 (f + g) (m(f ) + ε)2 = m2 (f ) + 2m(f )ε + ε 2 3ε2 for sufficiently small ε.
Definition 2.3. A Dobrakov submeasure is nonatomic, if to any f ∈ F with (f ) > 0 there exists g ∈ F such that g f and 0 < (g) < (f ). Definition 2.4. A Dobrakov submeasure has the Saks property (see [20]), if for any ε > 0 and any f ∈ F there exists an ε- partition of f, i.e. such elements f1 , f2 , . . . , fk ∈ F that ki=1 fi = f , and (fi ) < ε (i = 1, 2, . . . , k). Definition 2.5. A Dobrakov submeasure has the Darboux property, if to any f ∈ F with (f ) > 0 and any t ∈ (0, (f )) there exists g ∈ F such that g f and (g) = t.
3. Main result Theorem 3.1. For any Dobrakov submeasure the following three properties are equivalent: (i) is nonatomic; (ii) has the Saks property; (iii) has the Darboux property. Since the Darboux property evidently implies nonatomicity, the proof of theorem follows from the following two propositions. The main ideas of the proof are taken from [11]. Lemma 3.1. If a Dobrakov submeasure is nonatomic, and f ∈ F , (f ) > 0, then to any ε > 0 there exists h ∈ F such that h f and 0 < (h) < ε. Proof. Contrary, assume that there exists ε > 0 such that for any h ∈ F , h f either (h) ε or (h) = 0. Since is nonatomic, there exists h1 ∈ F , h1 f such that 0 < (h1 ) < (f ). By preceding (h1 ) ε. Consider f − h1 . Evidently 0 f − h1 f . By assumption 2 of Definition 2.1, f − h1 ∈ F . We want to prove that (f − h1 ) > 0. Put = 1/2((f ) − (h1 )) > 0. By (ii) of Definition 2.3 there exists > 0 such that (h1 + k) < (h1 ) +
whenever k ∈ F , (k) < . If (f − h1 ) = 0, then (put k = f − h1 ) (f ) = (h1 + f − h1 ) (h1 ) + = (h1 ) + 21 ((f ) − (h1 ))
= 21 ((f ) + (h1 )) < (f ), which is a contradiction. Hence (f − h1 ) > 0.
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We have proved that there exists h2 f − h we obtain a 1 with (h2 ) ε. Continuing the process ∞ ∞ sequence (hn )∞ of elements of F such that h f and (h ) ε. Put g = n n n n=1 i=n hi . Then n=1 gn ∈ F (n = 1, 2, . . .), gn 0. Since then (gn ) 0, there exists such n that (hn ) (gn ) < ε,
which is a contradiction.
Proposition 3.1. If a Dobrakov submeasure is nonatomic, then has the Saks property. Proof. Let g ∈ F , ε > 0. We want to construct an ε-partition of g. Put a1 = sup{(f ); f ∈ F , f g, (f ) < ε}. If a1 = 0, then (g) = 0 < ε and the proof is finished. If a1 > 0, then there exists g1 ∈ F , g1 g such that a1 < (g1 ) a1 . (g1 ) < ε, 2 If (g − g1 ) < ε the proof is complete. If not, put a2 = sup{(f ); f ∈ F , f g − g1 , (f ) < ε} and construct g2 ∈ F , g2 g − g1 such that (g2 ) < ε,
a2 < (g2 ) a2 . 2
By such a way a sequence (gn ) (possibly infinite) can be constructed such that an < (gn ) an = sup{(f ); f ∈ F , f g − g1 − · · · − gn−1 , (f ) < ε}. 2 Put hn = ∞ i=n gi . Then 0<
0<
an < (gn ) (hn ). 2
Since hn g, hn 0, we obtain (hn ) 0 and hence (1) limn→∞ a n = 0. Put f = g − ∞ i=1 gi . We want to prove that (f ) = 0. On the contrary let (f ) > 0. By Lemma 3.1 there exists h ∈ F , h f such that 0 < (h) < ε. Of course, hf = g −
∞ i=1
gi g −
k
gi (k = 1, 2, . . .).
i=1
Since, moreover, (h) < ε, we obtain (2) (h) ak (k = 1, 2, . . .). By (1) and (2) we conclude that (h) 0, which is a contradiction.
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Since ∞ i=n gi 0 there exists n0 such that ∞ gi < ε. i=n0
Consider the family ∞ A = g1 , g2 , . . . , gn0 −1 , gi , f . i=n0
Then (k) < ε for any k ∈ F , and moreover
∞ ∞ g1 + · · · + gn0 −1 + gi + g − gi = g, i=n0
hence A is an ε-partition of g.
i=1
Proposition 3.2. If a Dobrakov submeasure has the Saks property, then has also the Darboux property. Proof. Let f ∈ F , 0 < t < (f ). Let (εn ) be a sequence of real numbers, εn 0, εn < t. Because of the Saks property there exists an ε1 -partition {f1 , . . . , fk } of f. Since (f1 ) < ε1 < t, but (f1 + · · · + fk ) = (f ) > t, there exists l such that (f1 + · · · + fl ) < t, (f1 + · · · + fl + fl+1 ) t.
Put g1 = f1 + · · · + fl , h1 = f1 + · · · + fl + fl+1 . Then g1 h1 f, (g1 ) < t, (h1 ) t, (h1 − g1 ) = (fl+1 ) < ε1 .
Apply now the Saks property to the function h1 − g1 . There exist k1 , . . . , kn ∈ F such that h1 − g1 =
n
ki , (ki ) < ε2
(i = 1, . . . , n).
i=1
Again, since (g1 ) < t, (g1 + k1 + · · · + kn ) = (g1 + (h1 − g1 )) t, there exists u such that (g1 + k1 + · · · + ku ) < t, (g1 + k1 + · · · + ku + ku+1 ) t.
Put g2 = g1 + k1 + · · · + ku , h2 = g1 + k1 + · · · + ku + ku+1 . Then g1 g2 h2 h1 f, (g2 ) < t, (h2 ) t, (h2 − g2 ) = (ku+1 ) < ε2 .
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By this method we can obtain two sequences (gn ), (hn ) of elements of F such that g1 g2 · · · gn hn · · · h2 h1 f, (hn − gn ) < εn
(n = 1, 2, . . .).
Put g=
∞
gn .
n=1
Then g ∈ F , g f . We want to prove that (g) = t. First we shall show that the inequality (g) < t is impossible. If (g) < t, then we can put ε = t − (g) > 0. By (ii) there exists > 0 such that k + g ∈ F , (g + k) (g) + ε, whenever k ∈ F and (k) < . Since εn 0, there exists n0 such that (hn0 − gn0 ) εn0 < .
Then (hn0 ) = (gn0 + (hn0 − gn0 )) (g + (hn0 − gn0 )) < (g) + ε = t,
which is a contradiction. We have proved that (g) t. Let ε > 0. By (ii) there exists > 0 such that for any k ∈ F with (k) < we obtain that gn0 + k ∈ F and (gn0 + k) (gn0 ) + ε. Since (g − gn0 ) (hn0 − gn0 ) < εn0 < ,
we have (g) = (gn0 + (g − gn0 )) < (gn0 ) + ε < t + ε.
Since (g) < t + ε for any ε > 0, we have (g) t, and therefore (g) = t.
4. Concluding remarks The results present contributions in two directions. Firstly, they are new and in some sense surprising from the point of view of fuzzy sets theory: the fact that the range of a given mapping is a convex set does not depend on the additivity of the given mapping, but only on a very weak kind of subadditivity. Secondly, from the mathematical point of view one can recognize new methods of proofs based on the Dobrakov ideas. For example they can be rearranged for the MV-algebra domains [9]. Acknowledgements Supported by grant VEGA 1/9056/02.
B. Rieˇcan / Fuzzy Sets and Systems 151 (2005) 635 – 641
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References [1] D. Butnariu, E.P. Klement, Triangular Norm-Based Measures and Games with Fuzzy Coalitions, Kluwer, Dordrecht, 1993. [2] D. Butnariu, E.P. Klement, Triangular, norm-based measures, in: E. Pap (Ed.), Handbook on Measure Theory, Elsevier, Amsterdam, 2002, pp. 947–1010. ˇ [3] P. Cernek, Product of submeasures, Acta Math. Univ. Comenian 40–41 (1982) 301–308. ˇ [4] P. Cernek, The least bound of the additive measures and integrals, Zb. Rad. Prirod. Mat. Fak. Ser. Mat. 24 (1) (1994) 73–80. [5] N. Dinculeanu, Vector Measures, VEB, Berlin, 1966. [6] I. Dobrakov, On submeasures I, Dissertationes Math. 112 (1974) 1–35. [7] I. Dobrakov, J. Dobrakovová, On submeasures II, Math. Slovaca 30 (1980) 65–81. [8] P.R. Halmos, The range of a vector measure, Bull. Amer. Math. Soc. 54 (1948) 416–421. [9] M. Jureˇcková, F. Chovanec, On some properties of submeasures on MV-algebras, Math. Slovaca 54 (2004) 161–167. [10] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer, Dordrecht, 2000. [11] E.P. Klement, M. Navara, A characterization of tribes with respect to the Lukasiewicz t-norm, Czechoslovak Math. J. 47 (1997) 689–700. [12] V.M. Klimkin, M.G. Svistula, The Darboux property of non-additive set functions, Mat. Sb. 192 (2001) 41–50 (in Russian). [13] D. Maharam, Finitely additive measures on integers, Sankya Ser. A 38 (1976) 44–49. [14] A. Martelotti, Topological properties of the range of a group-valued finitely additive measure, J. Math. Anal. Appl. 110 (1985) 411–424. [15] R. Mesiar, M. Navara, Ts -tribes and Ts -measures, J. Math. Anal. Appl. 201 (1996) 91–102. [16] K. Musial, Absolute continuity and the range of group valued measure, Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys. 21 (1973) 105–113. [17] V. Olejˇcek, Darboux property of finitely additive measure on -ring, Math. Slovaca 27 (1977) 195–201. [18] E. Pap, Null-Additive Set Functions, Kluwer, Dordrecht, 1995. [19] B. Rieˇcan, T. Neubrunn, Integral, Measure, and Ordering, Kluwer, Dordrecht, 1997. [20] S. Saks, Addition to the note on some functionals, Trans. Amer. Math. Soc. 35 (1933) 967–974.