Lusin's theorem on fuzzy measure spaces - Semantic Scholar
Fuzzy Sets and Systems 146 (2004) 121 – 133 www.elsevier.com/locate/fss
Lusin’s theorem on fuzzy measure spaces Jun Lia;∗ , Masami Yasudab a b
Department of Applied Mathematics, Southeast University, Nanjing 210096, People’s Republic of China Department of Mathematics & Informatics, Faculty of Science, Chiba University, Chiba 263-8522, Japan Received 25 August 2002; received in revised form 5 May 2003; accepted 7 May 2003
Abstract In this paper, we show that weakly null-additive fuzzy measures on metric spaces possess regularity. Lusin’s theorem, which is well-known in classical measure theory, is generalized to fuzzy measure space by using the regularity and weakly null-additivity. A version of Egoro3’s theorem for the fuzzy measure de4ned on metric spaces is given. An application of Lusin’s theorem to approximation in the mean of measurable function on fuzzy measure spaces is presented. c 2003 Elsevier B.V. All rights reserved. Keywords: Non-additive measures; Fuzzy measure; Weakly null-additivity; Regularity; Lusin’s theorem; Approximation in the mean
1. Introduction The well-known Lusin’s theorem in classical measure theory is very important and useful for discussing the continuity and the approximation of measurable function on metric spaces [8]. Song and Li [9] investigated the regularity of null-additive fuzzy measure on metric spaces and showed Lusin’s theorem on fuzzy measure space under the null-additivity condition. These improved the previous results of Wu and Ha [11]. Further discussions for the regularity of fuzzy measures were made by Pap [7], Jiang et al. [2,3], and Wu and Wu [12]. In this paper, we shall use a weaker structural characteristic of fuzzy measures—weakly nulladditivity—to discuss the above-mentioned problems. Our goal is to prove the Lusin’s theorem on fuzzy measure space under the weakly null-additivity condition. The paper is organized as follows. In Section 2, a necessary and suCcient condition of weakly null-additivity of fuzzy measure is presented in Lemma 1. It constitutes the essential position in our discussion here. In Section 3,
This work was supported by the China Scholarship Council. Corresponding author. Tel.: +86-25-3792396; fax: +86-25-3792396. E-mail address: [email protected] (J. Li).
∗
c 2003 Elsevier B.V. All rights reserved. 0165-0114/$ - see front matter doi:10.1016/S0165-0114(03)00207-0
122
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
we prove that the weakly null-additivity implies regularity for a 4nite fuzzy measure de4ned on metric space. In Section 4, a version of Egoro3’s theorem for the fuzzy measure de4ned on metric spaces is given. In Section 5, by using the regularity and Egoro3’s theorem we shall prove that the well-known Lusin’s theorem remains valid for those weakly null-additive fuzzy measures de4ned on a metric space. These are improvements and generalizations of the earlier results of Song and Li [9]. Lastly , as an application of Lusin’s theorem, we shall describe the mean approximations of measurable function by continuous functions, or by polynomials, or by step functions in the sense of Sugeno and of Choquet integral, respectively. 2. Preliminaries Throughout this paper, we suppose that (X; ) is a metric space, and that O and C are the classes of all open and closed sets in (X; ), respectively, and B is Borel -algebra on X , i.e., it is the smallest -algebra containing O [1]. Unless stated otherwise all the subsets mentioned are supposed to belong to B. A set function : B → [0; +∞] is called a fuzzy measure, if it satis4es the following properties: (FM1) (∅) = 0; (FM2) A⊂B implies (A)6(B) (monotonicity); (FM3) A1 ⊂A2 ⊂ · · · implies limn→∞ (An ) = ( ∞ n=1 An ) (continuity from below); (FM4) A1 ⊃A2 ⊃ · · ·, and there exists n0 with (An0 )¡+∞ imply ∞ (continuity from above): An lim (An ) = n→∞
n=1
In this paper, we always assume that is a 4nite fuzzy measure on B, i.e., (X )¡∞. A fuzzy measure is called null-additive, if (E ∪ F) = (E) whenever E; F ∈B and (F) = 0; autocontinuous from above, if limn→+∞ (E ∪ Fn ) = (E) whenever E ∈B; {Fn }⊂B, and limn→+∞ (Fn ) = 0 [10]. Denition 1 (Wang and Klir [10]). is called weakly null-additive, if for any E; F ∈B, (E) = (F) = 0 ⇒ (E ∪ F) = 0: Obviously, the null-additivity of implies weakly null-additivity. If is autocontinuous from above, then it is null-additive [10], and hence it is weakly null-additive. Lemma 1. is weakly null-additive if and only if for any ¿0 and any double sequence {An(k) | n¿1; k¿1}⊂B satisfying An(k) Dn (k → ∞), (Dn ) = 0; n = 1; 2; : : : ; there exists a subsequence {An(kn ) } of {An(k) | n¿1; k¿1} such that ∞ An(kn ) ¡ (k1 ¡ k2 ¡ · · ·): n=1
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
123
Proof. Necessity: Suppose is weakly null-additive. Write D = ∞ n=1 Dn , then by using the continuity from below of , we have (D) = 0 and Dn ⊂D (n = 1; 2; : : :). Since for any 4xed n = 1; 2; : : : ; An(k) Dn as k → ∞, we have An(k) ∪ D Dn ∪ D = D
(k → ∞)
for any 4xed n = 1; 2; : : : : For given ¿0, using the continuity from above of fuzzy measures, we have limk →+∞ (A1(k) ∪ D) = (D) = 0, therefore there exists k1 such that (A1(k1 ) ∪ D)¡ 2 ; For this k1 , (A1(k1 ) ∪ A2(k) ) ∪ D (A1(k1 ) ∪ D2 ) ∪ D = A1(k1 ) ∪ D; as k → ∞. Therefore it follows, from the continuity from above of , that lim ((A1(k1 ) ∪ A2(k) ) ∪ D) = (A1(k1 ) ∪ D):
k →+∞
Thus there exists k2 (¿k1 ), such that
((A1(k1 ) ∪ A2(k2 ) ) ∪ D) ¡ : 2 Generally, there exist k1 ; k2 ; : : : ; km , such that
((A1(k1 ) ∪ A2(k2 ) ∪ · · · ∪ Am(km ) ) ∪ D) ¡ : 2 (kn ) ∞ Hence we obtain a sequence {kn }∞ n=1 of numbers and a sequence {An }n=1 of sets. By using the monotonicity and the continuity from below of , we have +∞ +∞
An(kn ) 6 An(kn ) ∪ D 6 ¡ : 2 n=1 n=1
Su>ciency: Let E; F ∈B and (E) = (F) = 0. We de4ne a double sequence {An(k) | n¿1; k¿1} of sets satisfying the following conditions: A1(k) = E, A2(k) = F; A3(k) = A4(k) = · · · = ∅; ∀k¿1 and let D1 = E; D2 = F; Dn = ∅; ∀n¿3. Then for any ¿0, by hypothesis, there exists a subsequence {An(kn ) } (kn ) such that ( ∞ n=1 An )¡ , that is (E ∪ F)¡ . Therefore (E ∪ F) = 0. This shows that is weakly null-additive. Remark 1. A weakly null-additive fuzzy measure may not be null-additive. In the following, a simple example indicates that the weakly null-additivity of fuzzy measure is really weaker than null-additivity and autocontinuity from above. Example 1. Let X = {a; b} and (X; ) be a metric space. Then B = ˝(X ). Put 1 if E = X; (E) = 12 if E = {b}; 0 if E = {a} or E = ∅:
124
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
Then is a fuzzy measure with weakly null-additivity. However is not null-additive and hence it is not autocontinuous from above either. In fact, ({a}) = 0, but ({a} ∪ {b}) = 1 = ({b}).
3. Regularity of fuzzy measure It is known that every probability measure P on a metric space is regular. Now we prove that this property is also enjoyed by those fuzzy measures with weakly null-additivity. Denition 2 (Wu and Ha [11]). is called regular if, for every A∈B and ¿0, there exist a closed set F and an open set G of X , such that F ⊂A⊂G and (G − F )¡ . Theorem 1. If is weakly null-additive, then is regular. Proof. Let E be the class of all set E ∈B such that for any ¿0, there exist a closed set F and an open set G satisfying F ⊂ E ⊂ G
and
(G − F ) ¡ :
To prove the theorem, it is suCcient to show that B⊂E. It is easy to verify that ∅∈E; X ∈E and E is closed under the formation of complements. We shall now prove that E is also closed under the formation of countable unions. Let {En }⊂E and ¿0 be given. From the de4nition of E and En ∈E, we know that for every n = 1; 2; : : : ; there (k) ∞ exist a sequence {Gn(k) }∞ k=1 of open sets and a sequence {Fn }k=1 of closed sets such that Fn(k) ⊂ En ⊂ Gn(k)
and
(Gn(k) − Fn(k) ) ¡
1 k
for k = 1; 2; : : : : Without loss of generality, we can assume that for 4xed n = 1; 2; : : : ; as k → ∞, (k) ∞ (k) {Gn(k) }∞ k=1 is decreasing and {Fn }k=1 is increasing. Therefore, for any 4xed n = 1; 2; : : : ; {Gn − Fn(k) }∞ k=1 is a decreasing sequence of sets with respect to k and as k → ∞ Gn(k)
−
Fn(k)
∞
(Gn(k) − Fn(k) ):
k=1
(k) (k) (k) (k) Dn as k → ∞ and noting that (Dn )6(Gn(k) − Denote Dn = ∞ k=1 (Gn − Fn ), then Gn − Fn (k) Fn )¡ k1 ; k = 1; 2; : : : ; we have (Dn ) = 0 (n = 1; 2; : : :). Applying Lemma 1 to the double sequence {Gn(k) − Fn(k) } and the sequence {Dn }∞ n=1 of sets, then for any given ¿0, there exists a subsequence {Gn(kn ) − Fn(kn ) } of {Gn(k) − Fn(k) } such that ∞ (Gn(kn ) − Fn(kn ) ) ¡ : n=1
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
125
Since ∞
Gn(kn )
n=1
−
N
Fn(kn )
n=1
∞
Gn(kn )
n=1
−
∞
Fn(kn )
n=1
(kn ) (kn ) ∞ as N → ∞, and noting that ∞ − ∞ ⊂ n=1 (Gn(kn ) − Fn(kn ) ), by the continuity from n=1 Gn n=1 Fn above and monotonicity of , we have ∞ ∞ N ∞ (kn ) (kn ) (kn ) (kn ) = ¡ : Gn − Fn Gn − Fn lim N →+∞
n=1
n=1
n=1
n=1
Therefore, there exists N0 such that ∞ N0 (kn ) (kn ) ¡ : Gn − Fn n=1
n=1
Denote G =
∞
Gn(kn )
and
F =
n=1
N0
Fn(kn )
n=1
then G is an open set, F is a closed set and F ⊂
∞
En ⊂ G
and
(G − F ) ¡ :
n=1
Therefore ∞ n=1 En ∈E. Thus we proved that E is a -algebra. To complete the proof, it is enough to show that E contains all the open sets of X . For any closed set F ∈C, we denote Gm = {x ∈X : (x; F)¡1=m} (m = 1; 2; : : :), where (x; F) is the distance of the set F from the point X , i.e. (x; F) = inf {(x; y): y ∈F}, then for every m = 1; 2; : : : ; Gm is open set. Noting that F is a closed set, we know Gm F (m → ∞). It is follows from Gm − F ∅ (m → ∞) that limm→∞ (Gm − F) = 0. Thus C ⊂E. Since E is closed under the formation of complements, we have O⊂E. This shows that E is a -algebra containing O. Therefore B⊂E. Corollary 1. If is weakly null-additive, then for any E ∈B, there exist a sequence {F (k) }∞ k=1 of (k) ∞ closed sets and a sequence {G }k=1 of open sets such that for every k = 1; 2; : : : ; F (k) ⊂E ⊂G (k) , (G (k) − E) ¡
1 k
and
(E − F (k) ) ¡
1 : k
Note 1: Observe that we can assume in Corollary 1 that the sequence {F (k) }∞ k=1 is increasing in k and the sequence {G (k) }∞ is decreasing in k. k=1
126
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
4. Egoro&’s theorem Egoro3’s theorem on fuzzy measure spaces was discussed in [4–6,10]. Now we show a version of the Egoro3’s theorem for the fuzzy measures de4ned on metric spaces. We assume that in this paper all functions considered are de4ned on X and are real-valued measurable with respect to B. For a 4nite fuzzy measure on B, we have obtained the following result [5]: Theorem 2 (Egoro3’s theorem). If {fn } converges to f almost everywhere on X , then for any ¿0 there exists X ∈B such that (X − X )¡ and {fn }n converges to f uniformly on X . The following corollary gives an alternative form of Egoro3’s theorem. Corollary 2. If {fn } converges to f almost ∞ everywhere on X , then there exists an increasing ∞ sequence {Xm }m=1 ⊂B such that (X − m=1 Xm ) = 0 and fn converges to f on Xm uniformly for any ?xed m = 1; 2; : : : : When is a weakly null-additive fuzzy measure on metric space, we can obtain a slightly stronger conclusion: Theorem 3. Let be weakly null-additive fuzzy measure on B. If {fn } converges to f almost everywhere on X , then for any ¿0 there exists a closed subset F ∈C such that (X − F )¡ and {fn }n converges to f uniformly on F . Proof. Since {fn } converges to f almost everywhere on X , by using Corollary 2 there exists an increasing sequence {Xm }∞ m=1 ⊂B such that fn converges ∞ to f on Xm uniformly for any 4xed m = 1; 2; : : : and (X − ∞ X ) = 0. Denote H = X − m=1 m m=1 Xm , then (H ) = 0. From Corollary 1, for every 4xed Xm (m = 1; 2; : : :), there exists a sequence {Fm(k) }∞ k=1 of closed sets satisfying Fm(k) ⊂Xm and (Xm − Fm(k) )¡1=k for any k = 1; 2; : : : : Without loss of generality, we can assume that for 4xed m = 1; 2; : : : ; {Xm − Fm(k) }∞ k=1 is decreasing (as k → ∞). Thus Xm − Fm(k)
∞
(Xm − Fm(k) )
k=1
as k → ∞. Write Dm = ( ∞ Fm(k) )) ∪ H (m = 1; 2; : : :), then (Xm − Fm(k) ) ∪ H Dm as k → ∞. k=1 (Xm −
∞ Noting that for any m = 1; 2; : : : ; ( k=1 (Xm −Fm(k) )) = limk →+∞ (Xm −Fm(k) ) = 0, and by the weakly
(k) null-additivity of , we get (Dm ) = (( ∞ k=1 (Xm −Fm )) ∪ H ) = 0 (m = 1; 2; : : :). Applying Lemma 1 to the double sequence {(Xm − Fm(k) ) ∪ H } of sets and the sequence {Dm }∞ m=1 of sets , then for any (km ) (k)
¿0, there exists a subsequence {(Xm − Fm ) ∪ H } of {(Xm − Fm ) ∪ H }, such that
∞
((Xm − Fm(km ) ) ∪ H )
m=1
¡ :
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
Since X −
∞
(km ) ∞ ⊂ m=1 m=1 Fm
X−
∞
Fm(km )
127
(Xm − Fm(k) ) ∪ H , we have
¡ :
m=1
(km ) On the other hand, from X − Nm=1 Fm(km ) X − ∞ m=1 Fm as N → ∞ and the continuity from N (km ) (km ) above of , we have limN →+∞ (X − m=1 Fm ) = (X − ∞ )¡ . Therefore there exists m=1 Fm N0 N0 such that (X − m=1 Fm(km ) )¡ . 0 0 Denote F = Nm=1 Fm(km ) , then F is a closed set, (X − F )¡ and from F ⊂ Nm=1 Xm , we know that {fn }n converges to f uniformly on F . 5. Lusin’s theorem In this section, we shall further generalize the well-known Lusin’s theorem in classical measure theory to fuzzy measure space by using the results obtained in Sections 2–4. Theorem 4 (Lusin’s theorem). Let be weakly null additive fuzzy measure on B. If f is a realvalued measurable function on X , then, for every ¿0, there exists a closed subset F ∈C such that f is continuous on F and (X − F )¡ . Proof. We prove the theorem stepwise in the following two situations. (a) Suppose that f is a simple function, i.e. f(x) = sn=1 ck En (x) (x ∈X ), where En (x) is the characteristic function of the set En and X = sn=1 En (a disjoint 4nite union). For every 4xed En (n = 1; 2; : : : ; s), by Corollary 1, there exists the sequence {Fn(k) }∞ k=1 of closed sets such that Fn(k) ⊂ En
and
(En − Fn(k) ) ¡
1 k
for any k = 1; 2; : : : : We may assume that {Fn(k) }∞ k=1 is increasing in k for each 4xed n, without any loss of generality. For any ¿0, applying Lemma 1 to the double sequence {En − Fn(k) } (n = 1; 2; : : : ; s; k = 1; 2; : : :) of sets, there exists a subsequence {En − Fn(kn ) } of {En − Fn(k) } such that s (kn ) (En − Fn ) ¡ : n=1
Put F =
s
(kn ) , n=1 Fn
then f is continuous on the closed subset F of X , and s s s En − Fn(kn ) 6 (En − Fn(kn ) ) ¡ : (X − F ) 6 n=1
n=1
n=1
(b) Let f be a real-valued measurable function. Then there exists a sequence {’n (x)}∞ n=1 of simple functions such that ’n → f (n → ∞) on X . By the result obtained in (a), for each simple function
128
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
’n and every k = 1; 2; : : : ; there exists closed set Xn(k) ⊂X such that ’n is continuous on Xn(k) and (X − Xn(k) )¡ k1 (k = 1; 2; : : :). There is no loss of generality in assuming the sequence {Xn(k) }∞ k=1 of k (i) closed sets is increasing with respect to k for any 4xed n (otherwise, we can take i=1 Xn instead of Xn(k) and noting that ’n is a simple function, it remains continuous on ki=1 Xn(i) ). Therefore
(k) X − Xn(k) ∞ k=1 (X − Xn ) as k → ∞, and thus, we have ∞ (X − Xn(k) ) = lim (X − Xn(k) ) = 0 (n = 1; 2; : : :): n→+∞
k=1
Now we consider the double sequence {X − Xn(k) | n¿1; k¿1} of sets. By using Lemma 1, for (m)
(k) every m (m = 1; 2; : : :), we may take a subsequence {X − Xn(kn ) }∞ n=1 of {X − Xn | n¿1; k¿1} such that ∞ (m) 1 (kn ) (X − Xn ) ¡ ; m n=1 (m)
(kn ) namely, (X − ∞ )¡1=m. Since the double sequence {X − Xn(k) | n¿1; k¿1} of sets is den=1 Xn creasing in k for 4xed n, without any loss of generality, we can assume that for 4xed n(n = 1; 2; : : :), (m)
(kn ) kn(1) ¡kn(2) ¡ · · · ¡kn(m) : : : : Write Hm = ∞ (m = 1; 2; : : :), then we obtain a sequence {Hm }∞ m=1 n=1 Xn of closed sets satisfying H1 ⊂H2 ⊂ · · · and (X − ∞ m=1 Hm )= lim n→+∞ (X − Hm )= 0. Noting that (m)
(m)
’n is continuous on Xn(kn ) and Hm ⊂Xn(kn ) (n = 1; 2; : : :), therefore for each Hm , ’n is continuous on Hm for every n = 1; 2; : : : : On the other hand, since ’n → f (n → ∞) on X, by Theorem 3, there existsan increasing sequence ∞ ∞ {Xm }∞ m=1 of closed sets satisfying X − Xm X − m=1 Xm (n →+∞), (X − m=1 Xm )= 0, and {’n } converges to f uniformly on closed set Xm for every m = 1; 2; : : : : Considering the sequence {(X − Hm ) ∪ (X − Xm )}∞ m=1 of sets, then, as m →+∞ ∞ ∞ Hm ∪ X − Xm : (X − Hm ) ∪ (X − Xm ) X − m=1
m=1
By using the continuity from above and weakly null-additivity of fuzzy measures, we have ∞ ∞ = 0: X− Hm ∪ X − Xm lim ((X − Hm ) ∪ (X − Xm )) = m→+∞
m=1
m=1
That is, limm→+∞ (X −Hm ∩ Xm ) = 0. Therefore, for given ¿0, we can take m0 such that (X − Hm0 ∩ Xm0 )¡ . Put F = Hm0 ∩ Xm0 , then F is a closed set and (X − F )¡ . Now we show that f is continuous on F . In fact, F ⊂Hm0 and ’n is continuous on Hm0 , therefore ’n is continuous on F for every n = 1; 2; : : : : Noting that {’n } converges to f on F uniformly, then f is continuous on F . Remark 2. Song and Li [9] have obtained the conclusions of Theorems 1, 3 and 4 under the null-additivity condition. As shown in Example 1, weekly null-additivity is really weaker than
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
129
null-additivity and autocontinuity from above. Therefore, Theorems 1, 3, and 4 in this paper are improvements of the related results in Song and Li [9] and, Wu and Ha [11].
6. Applications of Lusin’s theorem Now we present some applications of Lusin’s theorem to the mean approximation of measurable function by continuous functions, or by polynomials, or by step functions in the sense of Sugeno and of Choquet integral, respectively. Consider a nonnegative real-valued measurable function f on (X; B). The Sugeno( fuzzy) integral of f on X with respect to , denoted by (S) f d, is de4ned by
(S) f d = sup [$ ∧ ({x: f(x) ¿ $})]: 06$¡+∞
The Choquet integral of f on X with respect to , denoted by (C)
(C)
f d =
∞ 0
f d, is de4ned by
({x: f(x) ¿ t}) dt;
where the right side integral is Lebesgue integral. We say that a measurable function sequence {fn }n converges to f in fuzzy measure , and denote it by fn → f, if for any ¿0, limn→∞ ({x: |fn (x) − f(x)|¿ }) = 0. Theorem 5. Let be a weakly null-additive fuzzy measure on B. If f is a real-valued measurable function on X , then there exists a continuous function sequence { n }n on X such that n → f. Furthermore, if |f|6M , then | n |6M; n = 1; 2; : : : : Proof. For every n = 1; 2; : : : ; using Theorem 4 (Lusin’s theorem), we can obtain a closed subset Fn of X such that f is continuous on Fn and (X − Fn )¡ 1n . By Tietze’s extension theorem [8], for every n = 1; 2; : : : ; there exists continuous function n on X such that n (x) = f(x) for x ∈Fn , and if |f|6M , then | n |6M . Now we show that { n }n converges to f in fuzzy measure. In fact, for any ¿0, we have {x: | n (x) − f(x)|¿ }⊂X − Fn , and therefore ({x: | n (x) − f(x)|¿ })6(X − Fn )¡ 1n ; n = 1; 2; : : : : Thus we have limn→∞ ({x: | n (x) − f(x)|¿ }) = 0. The following result can be thought as to be the mean approximation theorem on fuzzy measure spaces (X; B; ). Theorem 6. Let be a weakly null-additive fuzzy measure on B. If f is a real-valued measurable function on X , then there exists a continuous function sequence { n }n on X such that
lim (S) | n − f| d = 0; n→+∞
130
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
Furthermore, if |f|6M , then | n |6M (n = 1; 2; : : :) and
lim (C) | n − f| d = 0: n→+∞
Proof. From Theorem 5, there exists a continuous function sequence { n }n on X such that By using Theorem 7.4 in [10], we can directly obtain limn→+∞ (S) | n − f| d = 0. If |f|6M , then from Theorem 5, | n |6M (n = 1; 2; : : :). Put gn (t) = ({x: | n (x) − f(x)| ¿ t});
n
→ f.
t ∈ [0; +∞)
a:e:
since n → f, we have gn (t) → 0 on [0; +∞) as n → ∞. Note that |gn (t)|6(X )¡∞, and gn (t) = 0 for any t¿2M (n = 1; 2; : : :). Applying the Bounded Convergence Theorem in Lebesgue integral theory [8] to the function sequence {gn (t)}n , we have
∞ 0
2M
gn (t)dt =
That is, limn→+∞ (C)
0
|
gn (t) dt → 0 n
(n → ∞):
− f| d = 0.
In the following, we discuss the mean approximation of measurable function either by polynomials or by step functions on fuzzy measure space (R1 ; B; ). Theorem 7. Let be a weakly null-additive fuzzy measure on B. If f is a real-valued measurable function on [a; b], then there exists a sequence {Pn }n of polynomials on [a; b] such that Pn → f. Furthermore, if |f|6M , then |Pn |6M + 1; n = 1; 2; : : : : Proof. Considering the problem on the reduced fuzzy measure space ([a; b]; [a; b] ∩ B; ), then we can from Theorem 5 obtain a continuous function sequence { n }n on [a; b] such that n → f on [a; b]. Therefore, there exists a subsequence { nk }k of { n }n , such that 1 1 ¡ ; x: | nk (x) − f(x)| ¿ 2k k for any k = 1; 2; : : : : Since nk is continuous function on [a; b] (k = 1; 2; : : :), by using Weierstrass’s theorem [8], for every k = 1; 2; : : : ; there exists a polynomial Pk on [a; b] such that for all x ∈[a; b] |Pk (x) −
nk (x)|
¡
1 : 2k
Thus, for every k = 1; 2; : : : ; we have 1 = ∅: x: |Pk (x) − nk (x)| ¿ 2k
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
Noting that 1 x: |Pk (x) − f(x)| ¿ ⊂ Ak ∪ Bk = x: | k where
1 x: |Pk (x) − nk (x)| ¿ 2k
Ak = and
Bk =
1 x: | nk (x) − f(x)| ¿ 2k
nk (x) − f(x)| ¿
1 2k
131
;
;
therefore we have 1 1 ¡ : x: |Pk (x) − f(x)| ¿ k k
Now we show that Pn → f on [a, b]. In fact, for any given ¿0, we take n0 such that 1=n0 ¡ , then n¿n0 , 1 ; {x: |Pn (x) − f(x)| ¿ } ⊂ x: |Pn (x) − f(x)| ¿ n and therefore,
({x: |Pn (x) − f(x)| ¿ }) 6
1 x: |Pn (x) − f(x)| ¿ n
1 ¡ ; n
where n¿n0 . This shows Pn → f. In the proof above, if |f|6M , then | nk |6M . Since for every Pk , |Pk (x) − x ∈[a; b], we have |Pn |6M + 1; n = 1; 2; : : : :
1 nk (x)|¡ 2k
for all
Theorem 8. Let be a weakly null-additive fuzzy measure on B. If f is a real-valued measurable function on [a; b], then there exists a sequence {Pn }n of polynomials on [a; b] such that
lim (S) |Pn − f| d = 0: n→+∞
Furthermore, if |f|6M , then |Pn |6M + 1 (n = 1; 2; : : :) and
lim (C) |Pn − f| d = 0: n→+∞
Proof. It is similar to the proof of Theorem 6.
132
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
Similarly, we can obtain the following result: Theorem 9. Let be a weakly null-additive fuzzy measure on B. If f is a real-valued measurable function on [a; b], then there exists a sequence {sn }n of step functions on [a; b] such that sn → f and
lim (S) |sn − f| d = 0; n→+∞
Furthermore, if |f| is Choquet integrable, i.e., (C) |f| d¡∞, then |sn | is also Choquet integrable and
lim (C) |sn − f| d = 0: n→+∞
Corollary 3. If is null-additive fuzzy measure on B, then the conclusions of Theorems 5–9 hold. 7. Concluding remarks We have proved Lusin’s theorem on 4nite fuzzy measure space under the weakly null-additivity condition. As we have seen, the weakly null-additivity, including its a necessary and suCcient condition presented in Lemma 1, and the regularity of fuzzy measures play important roles in our discussions. It should be pointed out that our discussion on the weakly null-additivity is nothing but suCcient, not necessary for Theorem 1, 4, 5 and 6. Compared with the null-additivity and autocontinuity, it is a weaker requirement, we need still further discussion. Example 2. Let X = {a; b} and (X; ) be a metric space. Then B = ˝(X ). Put 1 if E = X; (E) = 0 if E = X: Then fuzzy measure is not weakly null-additive. But is regular and any measurable function is continuous on X , and hence Lusin’s theorem holds on (X; B; ). We do not know whether the weakly null-additivity condition may be abandoned in our discussion. In our further research, we intend to address this issue and to investigate whether Lusin’s theorem remains valid on 4nite fuzzy measure spaces (X; B; ) without any additional condition as the Egoro3’s theorem we have proved in [5]. Acknowledgements The authors are grateful to the referees and the concerned Area Editor for their valuable suggestions to revise this paper.
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
133
References [1] P.R. Halmos, Measure Theory, Van Nostrand, New York, 1968. [2] Q. Jiang, H. Suzuki, Fuzzy measures on metric spaces, Fuzzy Sets and Systems 83 (1996) 99–106. [3] Q. Jiang, S. Wang, D. Ziou, A further investigation for fuzzy measures on metric spaces, Fuzzy Sets and Systems 105 (1999) 293–297. [4] J. Li, A note on the null-additivity of the fuzzy measure, Fuzzy Sets and Systems 125 (2002) 269–271. [5] J. Li, On Egoro3’s theorems on fuzzy measure space, Fuzzy Sets and Systems 135 (2003) 367–375. [6] J. Li, M. Yasuda, Q. Jiang, H. Suzuki, Z. Wang, G.J. Klir, Convergence of sequence of measurable functions on fuzzy measure space, Fuzzy Sets and Systems 87 (1997) 317–323. [7] E. Pap, Null-Additive Set Functions, Kluwer, Dordrecht, 1995. [8] H.L. Royden, Real Analysis, Macmillan, New York, 1966. [9] J. Song, J. Li, Regularity of null-additive fuzzy measure on metric spaces, Int. J. Gen. Systems, to appear. [10] Z. Wang, G.J. Klir, Fuzzy Measure Theory, Plenum, New York, 1992. [11] C. Wu, M. Ha, On the regularity of the fuzzy measure on metric fuzzy measure spaces, Fuzzy Sets and Systems 66 (1994) 373–379. [12] J. Wu, C. Wu, Fuzzy regular measures on topological spaces, Fuzzy Sets and Systems 119 (2001) 529–533.
Lusin’s theorem on fuzzy measure spaces Jun Lia;∗ , Masami Yasudab a b
Department of Applied Mathematics, Southeast University, Nanjing 210096, People’s Republic of China Department of Mathematics & Informatics, Faculty of Science, Chiba University, Chiba 263-8522, Japan Received 25 August 2002; received in revised form 5 May 2003; accepted 7 May 2003
Abstract In this paper, we show that weakly null-additive fuzzy measures on metric spaces possess regularity. Lusin’s theorem, which is well-known in classical measure theory, is generalized to fuzzy measure space by using the regularity and weakly null-additivity. A version of Egoro3’s theorem for the fuzzy measure de4ned on metric spaces is given. An application of Lusin’s theorem to approximation in the mean of measurable function on fuzzy measure spaces is presented. c 2003 Elsevier B.V. All rights reserved. Keywords: Non-additive measures; Fuzzy measure; Weakly null-additivity; Regularity; Lusin’s theorem; Approximation in the mean
1. Introduction The well-known Lusin’s theorem in classical measure theory is very important and useful for discussing the continuity and the approximation of measurable function on metric spaces [8]. Song and Li [9] investigated the regularity of null-additive fuzzy measure on metric spaces and showed Lusin’s theorem on fuzzy measure space under the null-additivity condition. These improved the previous results of Wu and Ha [11]. Further discussions for the regularity of fuzzy measures were made by Pap [7], Jiang et al. [2,3], and Wu and Wu [12]. In this paper, we shall use a weaker structural characteristic of fuzzy measures—weakly nulladditivity—to discuss the above-mentioned problems. Our goal is to prove the Lusin’s theorem on fuzzy measure space under the weakly null-additivity condition. The paper is organized as follows. In Section 2, a necessary and suCcient condition of weakly null-additivity of fuzzy measure is presented in Lemma 1. It constitutes the essential position in our discussion here. In Section 3,
This work was supported by the China Scholarship Council. Corresponding author. Tel.: +86-25-3792396; fax: +86-25-3792396. E-mail address: [email protected] (J. Li).
∗
c 2003 Elsevier B.V. All rights reserved. 0165-0114/$ - see front matter doi:10.1016/S0165-0114(03)00207-0
122
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
we prove that the weakly null-additivity implies regularity for a 4nite fuzzy measure de4ned on metric space. In Section 4, a version of Egoro3’s theorem for the fuzzy measure de4ned on metric spaces is given. In Section 5, by using the regularity and Egoro3’s theorem we shall prove that the well-known Lusin’s theorem remains valid for those weakly null-additive fuzzy measures de4ned on a metric space. These are improvements and generalizations of the earlier results of Song and Li [9]. Lastly , as an application of Lusin’s theorem, we shall describe the mean approximations of measurable function by continuous functions, or by polynomials, or by step functions in the sense of Sugeno and of Choquet integral, respectively. 2. Preliminaries Throughout this paper, we suppose that (X; ) is a metric space, and that O and C are the classes of all open and closed sets in (X; ), respectively, and B is Borel -algebra on X , i.e., it is the smallest -algebra containing O [1]. Unless stated otherwise all the subsets mentioned are supposed to belong to B. A set function : B → [0; +∞] is called a fuzzy measure, if it satis4es the following properties: (FM1) (∅) = 0; (FM2) A⊂B implies (A)6(B) (monotonicity); (FM3) A1 ⊂A2 ⊂ · · · implies limn→∞ (An ) = ( ∞ n=1 An ) (continuity from below); (FM4) A1 ⊃A2 ⊃ · · ·, and there exists n0 with (An0 )¡+∞ imply ∞ (continuity from above): An lim (An ) = n→∞
n=1
In this paper, we always assume that is a 4nite fuzzy measure on B, i.e., (X )¡∞. A fuzzy measure is called null-additive, if (E ∪ F) = (E) whenever E; F ∈B and (F) = 0; autocontinuous from above, if limn→+∞ (E ∪ Fn ) = (E) whenever E ∈B; {Fn }⊂B, and limn→+∞ (Fn ) = 0 [10]. Denition 1 (Wang and Klir [10]). is called weakly null-additive, if for any E; F ∈B, (E) = (F) = 0 ⇒ (E ∪ F) = 0: Obviously, the null-additivity of implies weakly null-additivity. If is autocontinuous from above, then it is null-additive [10], and hence it is weakly null-additive. Lemma 1. is weakly null-additive if and only if for any ¿0 and any double sequence {An(k) | n¿1; k¿1}⊂B satisfying An(k) Dn (k → ∞), (Dn ) = 0; n = 1; 2; : : : ; there exists a subsequence {An(kn ) } of {An(k) | n¿1; k¿1} such that ∞ An(kn ) ¡ (k1 ¡ k2 ¡ · · ·): n=1
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
123
Proof. Necessity: Suppose is weakly null-additive. Write D = ∞ n=1 Dn , then by using the continuity from below of , we have (D) = 0 and Dn ⊂D (n = 1; 2; : : :). Since for any 4xed n = 1; 2; : : : ; An(k) Dn as k → ∞, we have An(k) ∪ D Dn ∪ D = D
(k → ∞)
for any 4xed n = 1; 2; : : : : For given ¿0, using the continuity from above of fuzzy measures, we have limk →+∞ (A1(k) ∪ D) = (D) = 0, therefore there exists k1 such that (A1(k1 ) ∪ D)¡ 2 ; For this k1 , (A1(k1 ) ∪ A2(k) ) ∪ D (A1(k1 ) ∪ D2 ) ∪ D = A1(k1 ) ∪ D; as k → ∞. Therefore it follows, from the continuity from above of , that lim ((A1(k1 ) ∪ A2(k) ) ∪ D) = (A1(k1 ) ∪ D):
k →+∞
Thus there exists k2 (¿k1 ), such that
((A1(k1 ) ∪ A2(k2 ) ) ∪ D) ¡ : 2 Generally, there exist k1 ; k2 ; : : : ; km , such that
((A1(k1 ) ∪ A2(k2 ) ∪ · · · ∪ Am(km ) ) ∪ D) ¡ : 2 (kn ) ∞ Hence we obtain a sequence {kn }∞ n=1 of numbers and a sequence {An }n=1 of sets. By using the monotonicity and the continuity from below of , we have +∞ +∞
An(kn ) 6 An(kn ) ∪ D 6 ¡ : 2 n=1 n=1
Su>ciency: Let E; F ∈B and (E) = (F) = 0. We de4ne a double sequence {An(k) | n¿1; k¿1} of sets satisfying the following conditions: A1(k) = E, A2(k) = F; A3(k) = A4(k) = · · · = ∅; ∀k¿1 and let D1 = E; D2 = F; Dn = ∅; ∀n¿3. Then for any ¿0, by hypothesis, there exists a subsequence {An(kn ) } (kn ) such that ( ∞ n=1 An )¡ , that is (E ∪ F)¡ . Therefore (E ∪ F) = 0. This shows that is weakly null-additive. Remark 1. A weakly null-additive fuzzy measure may not be null-additive. In the following, a simple example indicates that the weakly null-additivity of fuzzy measure is really weaker than null-additivity and autocontinuity from above. Example 1. Let X = {a; b} and (X; ) be a metric space. Then B = ˝(X ). Put 1 if E = X; (E) = 12 if E = {b}; 0 if E = {a} or E = ∅:
124
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
Then is a fuzzy measure with weakly null-additivity. However is not null-additive and hence it is not autocontinuous from above either. In fact, ({a}) = 0, but ({a} ∪ {b}) = 1 = ({b}).
3. Regularity of fuzzy measure It is known that every probability measure P on a metric space is regular. Now we prove that this property is also enjoyed by those fuzzy measures with weakly null-additivity. Denition 2 (Wu and Ha [11]). is called regular if, for every A∈B and ¿0, there exist a closed set F and an open set G of X , such that F ⊂A⊂G and (G − F )¡ . Theorem 1. If is weakly null-additive, then is regular. Proof. Let E be the class of all set E ∈B such that for any ¿0, there exist a closed set F and an open set G satisfying F ⊂ E ⊂ G
and
(G − F ) ¡ :
To prove the theorem, it is suCcient to show that B⊂E. It is easy to verify that ∅∈E; X ∈E and E is closed under the formation of complements. We shall now prove that E is also closed under the formation of countable unions. Let {En }⊂E and ¿0 be given. From the de4nition of E and En ∈E, we know that for every n = 1; 2; : : : ; there (k) ∞ exist a sequence {Gn(k) }∞ k=1 of open sets and a sequence {Fn }k=1 of closed sets such that Fn(k) ⊂ En ⊂ Gn(k)
and
(Gn(k) − Fn(k) ) ¡
1 k
for k = 1; 2; : : : : Without loss of generality, we can assume that for 4xed n = 1; 2; : : : ; as k → ∞, (k) ∞ (k) {Gn(k) }∞ k=1 is decreasing and {Fn }k=1 is increasing. Therefore, for any 4xed n = 1; 2; : : : ; {Gn − Fn(k) }∞ k=1 is a decreasing sequence of sets with respect to k and as k → ∞ Gn(k)
−
Fn(k)
∞
(Gn(k) − Fn(k) ):
k=1
(k) (k) (k) (k) Dn as k → ∞ and noting that (Dn )6(Gn(k) − Denote Dn = ∞ k=1 (Gn − Fn ), then Gn − Fn (k) Fn )¡ k1 ; k = 1; 2; : : : ; we have (Dn ) = 0 (n = 1; 2; : : :). Applying Lemma 1 to the double sequence {Gn(k) − Fn(k) } and the sequence {Dn }∞ n=1 of sets, then for any given ¿0, there exists a subsequence {Gn(kn ) − Fn(kn ) } of {Gn(k) − Fn(k) } such that ∞ (Gn(kn ) − Fn(kn ) ) ¡ : n=1
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
125
Since ∞
Gn(kn )
n=1
−
N
Fn(kn )
n=1
∞
Gn(kn )
n=1
−
∞
Fn(kn )
n=1
(kn ) (kn ) ∞ as N → ∞, and noting that ∞ − ∞ ⊂ n=1 (Gn(kn ) − Fn(kn ) ), by the continuity from n=1 Gn n=1 Fn above and monotonicity of , we have ∞ ∞ N ∞ (kn ) (kn ) (kn ) (kn ) = ¡ : Gn − Fn Gn − Fn lim N →+∞
n=1
n=1
n=1
n=1
Therefore, there exists N0 such that ∞ N0 (kn ) (kn ) ¡ : Gn − Fn n=1
n=1
Denote G =
∞
Gn(kn )
and
F =
n=1
N0
Fn(kn )
n=1
then G is an open set, F is a closed set and F ⊂
∞
En ⊂ G
and
(G − F ) ¡ :
n=1
Therefore ∞ n=1 En ∈E. Thus we proved that E is a -algebra. To complete the proof, it is enough to show that E contains all the open sets of X . For any closed set F ∈C, we denote Gm = {x ∈X : (x; F)¡1=m} (m = 1; 2; : : :), where (x; F) is the distance of the set F from the point X , i.e. (x; F) = inf {(x; y): y ∈F}, then for every m = 1; 2; : : : ; Gm is open set. Noting that F is a closed set, we know Gm F (m → ∞). It is follows from Gm − F ∅ (m → ∞) that limm→∞ (Gm − F) = 0. Thus C ⊂E. Since E is closed under the formation of complements, we have O⊂E. This shows that E is a -algebra containing O. Therefore B⊂E. Corollary 1. If is weakly null-additive, then for any E ∈B, there exist a sequence {F (k) }∞ k=1 of (k) ∞ closed sets and a sequence {G }k=1 of open sets such that for every k = 1; 2; : : : ; F (k) ⊂E ⊂G (k) , (G (k) − E) ¡
1 k
and
(E − F (k) ) ¡
1 : k
Note 1: Observe that we can assume in Corollary 1 that the sequence {F (k) }∞ k=1 is increasing in k and the sequence {G (k) }∞ is decreasing in k. k=1
126
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
4. Egoro&’s theorem Egoro3’s theorem on fuzzy measure spaces was discussed in [4–6,10]. Now we show a version of the Egoro3’s theorem for the fuzzy measures de4ned on metric spaces. We assume that in this paper all functions considered are de4ned on X and are real-valued measurable with respect to B. For a 4nite fuzzy measure on B, we have obtained the following result [5]: Theorem 2 (Egoro3’s theorem). If {fn } converges to f almost everywhere on X , then for any ¿0 there exists X ∈B such that (X − X )¡ and {fn }n converges to f uniformly on X . The following corollary gives an alternative form of Egoro3’s theorem. Corollary 2. If {fn } converges to f almost ∞ everywhere on X , then there exists an increasing ∞ sequence {Xm }m=1 ⊂B such that (X − m=1 Xm ) = 0 and fn converges to f on Xm uniformly for any ?xed m = 1; 2; : : : : When is a weakly null-additive fuzzy measure on metric space, we can obtain a slightly stronger conclusion: Theorem 3. Let be weakly null-additive fuzzy measure on B. If {fn } converges to f almost everywhere on X , then for any ¿0 there exists a closed subset F ∈C such that (X − F )¡ and {fn }n converges to f uniformly on F . Proof. Since {fn } converges to f almost everywhere on X , by using Corollary 2 there exists an increasing sequence {Xm }∞ m=1 ⊂B such that fn converges ∞ to f on Xm uniformly for any 4xed m = 1; 2; : : : and (X − ∞ X ) = 0. Denote H = X − m=1 m m=1 Xm , then (H ) = 0. From Corollary 1, for every 4xed Xm (m = 1; 2; : : :), there exists a sequence {Fm(k) }∞ k=1 of closed sets satisfying Fm(k) ⊂Xm and (Xm − Fm(k) )¡1=k for any k = 1; 2; : : : : Without loss of generality, we can assume that for 4xed m = 1; 2; : : : ; {Xm − Fm(k) }∞ k=1 is decreasing (as k → ∞). Thus Xm − Fm(k)
∞
(Xm − Fm(k) )
k=1
as k → ∞. Write Dm = ( ∞ Fm(k) )) ∪ H (m = 1; 2; : : :), then (Xm − Fm(k) ) ∪ H Dm as k → ∞. k=1 (Xm −
∞ Noting that for any m = 1; 2; : : : ; ( k=1 (Xm −Fm(k) )) = limk →+∞ (Xm −Fm(k) ) = 0, and by the weakly
(k) null-additivity of , we get (Dm ) = (( ∞ k=1 (Xm −Fm )) ∪ H ) = 0 (m = 1; 2; : : :). Applying Lemma 1 to the double sequence {(Xm − Fm(k) ) ∪ H } of sets and the sequence {Dm }∞ m=1 of sets , then for any (km ) (k)
¿0, there exists a subsequence {(Xm − Fm ) ∪ H } of {(Xm − Fm ) ∪ H }, such that
∞
((Xm − Fm(km ) ) ∪ H )
m=1
¡ :
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
Since X −
∞
(km ) ∞ ⊂ m=1 m=1 Fm
X−
∞
Fm(km )
127
(Xm − Fm(k) ) ∪ H , we have
¡ :
m=1
(km ) On the other hand, from X − Nm=1 Fm(km ) X − ∞ m=1 Fm as N → ∞ and the continuity from N (km ) (km ) above of , we have limN →+∞ (X − m=1 Fm ) = (X − ∞ )¡ . Therefore there exists m=1 Fm N0 N0 such that (X − m=1 Fm(km ) )¡ . 0 0 Denote F = Nm=1 Fm(km ) , then F is a closed set, (X − F )¡ and from F ⊂ Nm=1 Xm , we know that {fn }n converges to f uniformly on F . 5. Lusin’s theorem In this section, we shall further generalize the well-known Lusin’s theorem in classical measure theory to fuzzy measure space by using the results obtained in Sections 2–4. Theorem 4 (Lusin’s theorem). Let be weakly null additive fuzzy measure on B. If f is a realvalued measurable function on X , then, for every ¿0, there exists a closed subset F ∈C such that f is continuous on F and (X − F )¡ . Proof. We prove the theorem stepwise in the following two situations. (a) Suppose that f is a simple function, i.e. f(x) = sn=1 ck En (x) (x ∈X ), where En (x) is the characteristic function of the set En and X = sn=1 En (a disjoint 4nite union). For every 4xed En (n = 1; 2; : : : ; s), by Corollary 1, there exists the sequence {Fn(k) }∞ k=1 of closed sets such that Fn(k) ⊂ En
and
(En − Fn(k) ) ¡
1 k
for any k = 1; 2; : : : : We may assume that {Fn(k) }∞ k=1 is increasing in k for each 4xed n, without any loss of generality. For any ¿0, applying Lemma 1 to the double sequence {En − Fn(k) } (n = 1; 2; : : : ; s; k = 1; 2; : : :) of sets, there exists a subsequence {En − Fn(kn ) } of {En − Fn(k) } such that s (kn ) (En − Fn ) ¡ : n=1
Put F =
s
(kn ) , n=1 Fn
then f is continuous on the closed subset F of X , and s s s En − Fn(kn ) 6 (En − Fn(kn ) ) ¡ : (X − F ) 6 n=1
n=1
n=1
(b) Let f be a real-valued measurable function. Then there exists a sequence {’n (x)}∞ n=1 of simple functions such that ’n → f (n → ∞) on X . By the result obtained in (a), for each simple function
128
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
’n and every k = 1; 2; : : : ; there exists closed set Xn(k) ⊂X such that ’n is continuous on Xn(k) and (X − Xn(k) )¡ k1 (k = 1; 2; : : :). There is no loss of generality in assuming the sequence {Xn(k) }∞ k=1 of k (i) closed sets is increasing with respect to k for any 4xed n (otherwise, we can take i=1 Xn instead of Xn(k) and noting that ’n is a simple function, it remains continuous on ki=1 Xn(i) ). Therefore
(k) X − Xn(k) ∞ k=1 (X − Xn ) as k → ∞, and thus, we have ∞ (X − Xn(k) ) = lim (X − Xn(k) ) = 0 (n = 1; 2; : : :): n→+∞
k=1
Now we consider the double sequence {X − Xn(k) | n¿1; k¿1} of sets. By using Lemma 1, for (m)
(k) every m (m = 1; 2; : : :), we may take a subsequence {X − Xn(kn ) }∞ n=1 of {X − Xn | n¿1; k¿1} such that ∞ (m) 1 (kn ) (X − Xn ) ¡ ; m n=1 (m)
(kn ) namely, (X − ∞ )¡1=m. Since the double sequence {X − Xn(k) | n¿1; k¿1} of sets is den=1 Xn creasing in k for 4xed n, without any loss of generality, we can assume that for 4xed n(n = 1; 2; : : :), (m)
(kn ) kn(1) ¡kn(2) ¡ · · · ¡kn(m) : : : : Write Hm = ∞ (m = 1; 2; : : :), then we obtain a sequence {Hm }∞ m=1 n=1 Xn of closed sets satisfying H1 ⊂H2 ⊂ · · · and (X − ∞ m=1 Hm )= lim n→+∞ (X − Hm )= 0. Noting that (m)
(m)
’n is continuous on Xn(kn ) and Hm ⊂Xn(kn ) (n = 1; 2; : : :), therefore for each Hm , ’n is continuous on Hm for every n = 1; 2; : : : : On the other hand, since ’n → f (n → ∞) on X, by Theorem 3, there existsan increasing sequence ∞ ∞ {Xm }∞ m=1 of closed sets satisfying X − Xm X − m=1 Xm (n →+∞), (X − m=1 Xm )= 0, and {’n } converges to f uniformly on closed set Xm for every m = 1; 2; : : : : Considering the sequence {(X − Hm ) ∪ (X − Xm )}∞ m=1 of sets, then, as m →+∞ ∞ ∞ Hm ∪ X − Xm : (X − Hm ) ∪ (X − Xm ) X − m=1
m=1
By using the continuity from above and weakly null-additivity of fuzzy measures, we have ∞ ∞ = 0: X− Hm ∪ X − Xm lim ((X − Hm ) ∪ (X − Xm )) = m→+∞
m=1
m=1
That is, limm→+∞ (X −Hm ∩ Xm ) = 0. Therefore, for given ¿0, we can take m0 such that (X − Hm0 ∩ Xm0 )¡ . Put F = Hm0 ∩ Xm0 , then F is a closed set and (X − F )¡ . Now we show that f is continuous on F . In fact, F ⊂Hm0 and ’n is continuous on Hm0 , therefore ’n is continuous on F for every n = 1; 2; : : : : Noting that {’n } converges to f on F uniformly, then f is continuous on F . Remark 2. Song and Li [9] have obtained the conclusions of Theorems 1, 3 and 4 under the null-additivity condition. As shown in Example 1, weekly null-additivity is really weaker than
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
129
null-additivity and autocontinuity from above. Therefore, Theorems 1, 3, and 4 in this paper are improvements of the related results in Song and Li [9] and, Wu and Ha [11].
6. Applications of Lusin’s theorem Now we present some applications of Lusin’s theorem to the mean approximation of measurable function by continuous functions, or by polynomials, or by step functions in the sense of Sugeno and of Choquet integral, respectively. Consider a nonnegative real-valued measurable function f on (X; B). The Sugeno( fuzzy) integral of f on X with respect to , denoted by (S) f d, is de4ned by
(S) f d = sup [$ ∧ ({x: f(x) ¿ $})]: 06$¡+∞
The Choquet integral of f on X with respect to , denoted by (C)
(C)
f d =
∞ 0
f d, is de4ned by
({x: f(x) ¿ t}) dt;
where the right side integral is Lebesgue integral. We say that a measurable function sequence {fn }n converges to f in fuzzy measure , and denote it by fn → f, if for any ¿0, limn→∞ ({x: |fn (x) − f(x)|¿ }) = 0. Theorem 5. Let be a weakly null-additive fuzzy measure on B. If f is a real-valued measurable function on X , then there exists a continuous function sequence { n }n on X such that n → f. Furthermore, if |f|6M , then | n |6M; n = 1; 2; : : : : Proof. For every n = 1; 2; : : : ; using Theorem 4 (Lusin’s theorem), we can obtain a closed subset Fn of X such that f is continuous on Fn and (X − Fn )¡ 1n . By Tietze’s extension theorem [8], for every n = 1; 2; : : : ; there exists continuous function n on X such that n (x) = f(x) for x ∈Fn , and if |f|6M , then | n |6M . Now we show that { n }n converges to f in fuzzy measure. In fact, for any ¿0, we have {x: | n (x) − f(x)|¿ }⊂X − Fn , and therefore ({x: | n (x) − f(x)|¿ })6(X − Fn )¡ 1n ; n = 1; 2; : : : : Thus we have limn→∞ ({x: | n (x) − f(x)|¿ }) = 0. The following result can be thought as to be the mean approximation theorem on fuzzy measure spaces (X; B; ). Theorem 6. Let be a weakly null-additive fuzzy measure on B. If f is a real-valued measurable function on X , then there exists a continuous function sequence { n }n on X such that
lim (S) | n − f| d = 0; n→+∞
130
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
Furthermore, if |f|6M , then | n |6M (n = 1; 2; : : :) and
lim (C) | n − f| d = 0: n→+∞
Proof. From Theorem 5, there exists a continuous function sequence { n }n on X such that By using Theorem 7.4 in [10], we can directly obtain limn→+∞ (S) | n − f| d = 0. If |f|6M , then from Theorem 5, | n |6M (n = 1; 2; : : :). Put gn (t) = ({x: | n (x) − f(x)| ¿ t});
n
→ f.
t ∈ [0; +∞)
a:e:
since n → f, we have gn (t) → 0 on [0; +∞) as n → ∞. Note that |gn (t)|6(X )¡∞, and gn (t) = 0 for any t¿2M (n = 1; 2; : : :). Applying the Bounded Convergence Theorem in Lebesgue integral theory [8] to the function sequence {gn (t)}n , we have
∞ 0
2M
gn (t)dt =
That is, limn→+∞ (C)
0
|
gn (t) dt → 0 n
(n → ∞):
− f| d = 0.
In the following, we discuss the mean approximation of measurable function either by polynomials or by step functions on fuzzy measure space (R1 ; B; ). Theorem 7. Let be a weakly null-additive fuzzy measure on B. If f is a real-valued measurable function on [a; b], then there exists a sequence {Pn }n of polynomials on [a; b] such that Pn → f. Furthermore, if |f|6M , then |Pn |6M + 1; n = 1; 2; : : : : Proof. Considering the problem on the reduced fuzzy measure space ([a; b]; [a; b] ∩ B; ), then we can from Theorem 5 obtain a continuous function sequence { n }n on [a; b] such that n → f on [a; b]. Therefore, there exists a subsequence { nk }k of { n }n , such that 1 1 ¡ ; x: | nk (x) − f(x)| ¿ 2k k for any k = 1; 2; : : : : Since nk is continuous function on [a; b] (k = 1; 2; : : :), by using Weierstrass’s theorem [8], for every k = 1; 2; : : : ; there exists a polynomial Pk on [a; b] such that for all x ∈[a; b] |Pk (x) −
nk (x)|
¡
1 : 2k
Thus, for every k = 1; 2; : : : ; we have 1 = ∅: x: |Pk (x) − nk (x)| ¿ 2k
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
Noting that 1 x: |Pk (x) − f(x)| ¿ ⊂ Ak ∪ Bk = x: | k where
1 x: |Pk (x) − nk (x)| ¿ 2k
Ak = and
Bk =
1 x: | nk (x) − f(x)| ¿ 2k
nk (x) − f(x)| ¿
1 2k
131
;
;
therefore we have 1 1 ¡ : x: |Pk (x) − f(x)| ¿ k k
Now we show that Pn → f on [a, b]. In fact, for any given ¿0, we take n0 such that 1=n0 ¡ , then n¿n0 , 1 ; {x: |Pn (x) − f(x)| ¿ } ⊂ x: |Pn (x) − f(x)| ¿ n and therefore,
({x: |Pn (x) − f(x)| ¿ }) 6
1 x: |Pn (x) − f(x)| ¿ n
1 ¡ ; n
where n¿n0 . This shows Pn → f. In the proof above, if |f|6M , then | nk |6M . Since for every Pk , |Pk (x) − x ∈[a; b], we have |Pn |6M + 1; n = 1; 2; : : : :
1 nk (x)|¡ 2k
for all
Theorem 8. Let be a weakly null-additive fuzzy measure on B. If f is a real-valued measurable function on [a; b], then there exists a sequence {Pn }n of polynomials on [a; b] such that
lim (S) |Pn − f| d = 0: n→+∞
Furthermore, if |f|6M , then |Pn |6M + 1 (n = 1; 2; : : :) and
lim (C) |Pn − f| d = 0: n→+∞
Proof. It is similar to the proof of Theorem 6.
132
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
Similarly, we can obtain the following result: Theorem 9. Let be a weakly null-additive fuzzy measure on B. If f is a real-valued measurable function on [a; b], then there exists a sequence {sn }n of step functions on [a; b] such that sn → f and
lim (S) |sn − f| d = 0; n→+∞
Furthermore, if |f| is Choquet integrable, i.e., (C) |f| d¡∞, then |sn | is also Choquet integrable and
lim (C) |sn − f| d = 0: n→+∞
Corollary 3. If is null-additive fuzzy measure on B, then the conclusions of Theorems 5–9 hold. 7. Concluding remarks We have proved Lusin’s theorem on 4nite fuzzy measure space under the weakly null-additivity condition. As we have seen, the weakly null-additivity, including its a necessary and suCcient condition presented in Lemma 1, and the regularity of fuzzy measures play important roles in our discussions. It should be pointed out that our discussion on the weakly null-additivity is nothing but suCcient, not necessary for Theorem 1, 4, 5 and 6. Compared with the null-additivity and autocontinuity, it is a weaker requirement, we need still further discussion. Example 2. Let X = {a; b} and (X; ) be a metric space. Then B = ˝(X ). Put 1 if E = X; (E) = 0 if E = X: Then fuzzy measure is not weakly null-additive. But is regular and any measurable function is continuous on X , and hence Lusin’s theorem holds on (X; B; ). We do not know whether the weakly null-additivity condition may be abandoned in our discussion. In our further research, we intend to address this issue and to investigate whether Lusin’s theorem remains valid on 4nite fuzzy measure spaces (X; B; ) without any additional condition as the Egoro3’s theorem we have proved in [5]. Acknowledgements The authors are grateful to the referees and the concerned Area Editor for their valuable suggestions to revise this paper.
J. Li, M. Yasuda / Fuzzy Sets and Systems 146 (2004) 121 – 133
133
References [1] P.R. Halmos, Measure Theory, Van Nostrand, New York, 1968. [2] Q. Jiang, H. Suzuki, Fuzzy measures on metric spaces, Fuzzy Sets and Systems 83 (1996) 99–106. [3] Q. Jiang, S. Wang, D. Ziou, A further investigation for fuzzy measures on metric spaces, Fuzzy Sets and Systems 105 (1999) 293–297. [4] J. Li, A note on the null-additivity of the fuzzy measure, Fuzzy Sets and Systems 125 (2002) 269–271. [5] J. Li, On Egoro3’s theorems on fuzzy measure space, Fuzzy Sets and Systems 135 (2003) 367–375. [6] J. Li, M. Yasuda, Q. Jiang, H. Suzuki, Z. Wang, G.J. Klir, Convergence of sequence of measurable functions on fuzzy measure space, Fuzzy Sets and Systems 87 (1997) 317–323. [7] E. Pap, Null-Additive Set Functions, Kluwer, Dordrecht, 1995. [8] H.L. Royden, Real Analysis, Macmillan, New York, 1966. [9] J. Song, J. Li, Regularity of null-additive fuzzy measure on metric spaces, Int. J. Gen. Systems, to appear. [10] Z. Wang, G.J. Klir, Fuzzy Measure Theory, Plenum, New York, 1992. [11] C. Wu, M. Ha, On the regularity of the fuzzy measure on metric fuzzy measure spaces, Fuzzy Sets and Systems 66 (1994) 373–379. [12] J. Wu, C. Wu, Fuzzy regular measures on topological spaces, Fuzzy Sets and Systems 119 (2001) 529–533.
