On the convergence theorems of generalized fuzzy integral sequence

Fuzzy Sets and Systems 124 (2001) 117–123

www.elsevier.com/locate/fss

On the convergence theorems of generalized fuzzy integral sequence  Jin-Xuan Fang Department of Mathematics, Nanjing Normal University, Nanjing Jiangsu, 210097, People’s Republic of China Received 8 December 1998; received in revised form 11 May 1999; accepted 27 March 2000

Abstract In this paper, we give a counterexample showing that two convergence theorems of generalized fuzzy integrals given by c 2001 Elsevier Wu et al. (Fuzzy Sets and Systems 70 (1995) 75 –87) is not valid and modify them to the right forms.
Science B.V. All rights reserved. Keywords: Generalized fuzzy integral; (S) fuzzy integral; (N ) fuzzy integral; Convergence theorem

1. Introduction and preliminaries In [3], Wu, Wang and Ma introduced the concept of generalized fuzzy integral (for short, (G) fuzzy integral) by means of the so-called generalized triangular norm. In [4], Wu, Ma, Song and Zhang investigated the convergence theorems of (G) fuzzy integrals. In this paper, we =rst point out that Theorems 7:1 and 7:2 in [4] are incorrect by a counterexample. Next, we prove several lemmas. Finally, we give three convergence theorems of (G) fuzzy integrals, which not only rectify and improve Theorems 7:1, 7:2 and 7:3 in [4] but also generalize the corresponding results of Wang and Klir [2]. We essentially use the same notations as in [3,4]. Let (X; A; ) be a fuzzy measure space. M+

denotes the set of nonnegative measurable func tions on X . Let {fn } ⊂ M+ and f ∈ M+ , fn → f denotes that {fn } converges to f in the fuzzy measure  on X . We write N ( f) = {x ∈ X : f(x)¿ } and N H (f) = {x ∈ X : f(x)¿ }. N denotes the set of natural numbers. Denition 1 (Wu et al. [3]). Let D = [0; ∞]2 \{(0; ∞); (∞; 0)}. The mapping S : D → [0; ∞] is said to be a generalized triangular norm if it satis=es the following conditions: (1) S[0; x] = 0 for all x ∈ [0; ∞) and there exists an e ∈ (0; ∞] such that S[x; e] = x for each x ∈ [0; ∞]. e is said to be the identity of S; (2) S[x; y] = S[y; x] for all (x; y) ∈ D; (3) S[a; b]6S[c; d] whenever a6c, b6d; (4) If {(xn ; yn )} ⊂ D, (x; y) ∈ D, xn  x and yn y, then S[xn ; yn ] → S[x; y].



Project supported by the National Natural Science Foundation of Educational Committee of Jiangsu Province of China. E-mail address: [email protected] (J.-X. Fang).

Remark 1. Obviously, if S is a generalized triangular norm, then [∞; ∞] = ∞.

c 2001 Elsevier Science B.V. All rights reserved. 0165-0114/01/$ - see front matter
PII: S 0 1 6 5 - 0 1 1 4 ( 0 0 ) 0 0 0 5 5 - 5

118

J.-X. Fang / Fuzzy Sets and Systems 124 (2001) 117–123

Remark 2. S1 [x; y] = min{x; y}, S2 [x; y] = kxy (k¿ 0) are the generalized triangular norms. It is easy to see that the identities of S1 and S2 are ∞ and 1=k, respectively. Let (X; A; ) be a fuzzy measure space, S a generalized triangular norm, A ∈ A
and ’ be a nonnegn ative simple function, i.e., ’ = i=1 i · Ai , where n A = i=1 Ai ; Ai ∩ Aj = ∅, i = j (i = j), i ¿0, Ai ∈ A (i = 1; 2; : : : ; n) and Ai is the characterization function of Ai . We de=ne QA (’) = sup S[ i ; (A ∩ Ai )]: 16i6n

Denition 2 (Wu et al. [3]). Let S be a generalized triangular norm, A ∈ A and f∈M+ . Then the generalized fuzzy integral (for short, (G) fuzzy integral) of f over A is de=ned by  f d = sup QA (’); A

0¡’6f

where ’ is a nonnegative simple function. ’¿0 means that ’¿0 and ’ ≡ 0. We provide sup  = 0. It is notable that the (G) fuzzy integral has the following expression. Proposition 1 (Wu et al. [3]). Let A ∈ A and f ∈ M+ . Then  f d = sup S[ ; (A ∩ N (f))] A

¿0

= sup S[ ; (A ∩ N H(f))]: ¿0

(1.1)

Remark. The (S) fuzzy integrals de=ned by Sugeno [1] and the (N ) fuzzy integrals de=ned by Zhao [5] are the special kinds of (G) fuzzy integrals; and the corresponding generalized triangular norms are S[x; y] = min{x; y} and S[x; y] = xy, respectively. Denition 3. Let (X; A; ) be a fuzzy measure space, E ∈ A and f : E → [− ∞; ∞]. We say that f is almost everywhere =nite on E, if there exists a set A ∈ A with (A) = 0 such that |f(x)|¡∞ for each x ∈ E\A. 2. A counterexample

Theorem A. (Wu et al. [4, Theorem 7:1]) Let (X; A; ) be a fuzzy measure space. Then whenever  A ∈ A and fn → f on A;  lim

n→∞

A

A

f d;

if and only if (1)  is autocontinuous; (2) limn→∞ S[1=n; ∞] = 0 or inf (A)¿0; A∈A (A)¿0; where {fn } ⊂ M+ ; f ∈ M+ and it is almost everywhere 7nite. Theorem B (Wu et al. [4, Theorem 7:2]). Let  be a  null-additive. Then fn → f implies that  lim

n→∞

|fn − f| d = 0;

if and only if limn→∞ S[1=n; ∞] = 0 or inf (A)¿0; A∈A (A)¿0; where {fn } ⊂ M+ ; f ∈ M+ and it is almost everywhere 7nite. We now give a counterexample to Theorems A and B as follows: Example. Let X = {1; 2; : : :}, A = P(X ). We de=ne (A) = |A| for each A ∈ P(X ); where |A| denotes the number of elements in A (when A is an in=nite set, we provide |A| = ∞). Obviously,  is a fuzzy measure on (X; A), it is autocontinuous (certainly, null-additive) and satis=es inf (A)¿0; A∈A (A)¿0. Take S[a; b] = ab, fn (x) ≡ (1=n) (n = 1; 2; : : :) and f(x) ≡ 0, then it is  easy to see that fn → f. However, 

 X

In [4], the authors have given the following two theorems.

 fn d =

|fn − f| d =

X

fn d = S[1=n; (X )]

= S[1=n; ∞] = ∞

J.-X. Fang / Fuzzy Sets and Systems 124 (2001) 117–123

and

 X

f d = 0. Consequently, we have

 lim

n→∞

 lim

n→∞

 X

X

fn d =

X

f d;

|fn − f| d = 0:

This shows that Theorems A and B are not true.

Proof. By (3), ∈ (0; 0 ), we can take a natural number m0 such that 0 − 1=m0 ¿ . Thus, for all m¿m0 we ∞have (N 0 −1=m (f))6(N (f))¡∞. Noting that m=m0 N 0 −1=m (f) = N H0 (f), by the continuity from above of , we have lim (N 0 −1=m (f)) = (N H0 (f)):

m→∞

Besides, it is not diMcult to prove that S[ 0 ; (N 0 −1=m (f))]¿M

3. Several lemmas To establish the correct convergence theorems for (G) fuzzy integral sequence, the following lemmas are helpful. Lemma 1. Every sequence {an } of nonnegative real numbers has a monotone subsequence {ani } that converges to a real number a or ∞. The proof is elementary, and is omitted. Lemma 2. Let S[·; ·] be a generalized triangular norm, and {an } a sequence of positive real numbers that converges to 0. Then S[an ; ∞] → 0 if and only if S[1=n; ∞] → 0. Proof. We only prove the necessity, the suMciency may be proved in a similar way. Suppose S[an ; ∞] → 0. Then for any given ¿0, there is an n0 such that S[an0 ; ∞]¡. Since 1=n → 0 and an0 ¿0, there exists an N ¿n0 such that 1=n¡an0 for all n¿N . By the monotone property of generalized triangular norm, we have S[1=n; ∞]6S[an0 ; ∞]¡ for all n¿N . Therefore S[1=n; ∞] → 0. Lemma 3. Let (X; A; ) be a fuzzy measure space;  be autocontinuous from above; {an } a monotone sequence of positive numbers and n → 0 ∈ (0; ∞). If {fn } ⊂ M+ ; f ∈ M+ and the following conditions are satis7ed:  (1) fn → f on X ; (2) S[ n ; (N n (fn ))]¿M (n = 1; 2; : : :); where M is a nonnegative real number; (3) there exists ∈ (0; 0 ) such that (N (f))¡∞. Then S[ 0 ; (N H0 (f))]¿M .

119

for all m¿m0 :

(3.1)

In fact, if n  0 , then for all m¿m0 we have n ¿ 0 ¿ 0 − 1=m¿ . Noting that N 0 (fn ) ⊂  N 0 −1=m (f) ∪ N1=m (|fn − f|), by that fn → f and the autocontinuity from above of , for any given ¿0 and above m¿m0 there exists an n0 such that (N 0 (fn )) ¡ (N 0 −1=m (f)) + 

for all n¿n0 :

Thus, by (2) we have M ¡ S[ n ; (N n (fn ))] 6S[ n ; (N 0 (fn ))] 6S[ n ; (N 0 −1=m (f)) + ]

for all n¿n0 :

(3.2)

Letting n → ∞, it follows from (3.2) that M 6 S[ 0 ; (N 0 −1=m (f)) + ] for all m¿m0 . Again letting   0 we get (3.1). If n 0 , then for any m¿m0 , there is an n0 such that 0 ¿ n ¿ 0 − 1=2m¿ for all n¿n0 . Thus, for all n¿n0 we have M ¡ S[ n ; (N n (fn ))]6S[ 0 ; (N 0 −1=2m (fn ))]: Noting that N 0 −1=2m (fn ) ⊂ N 0 −1=m (f) ∪N1=2m (|fn − f|), by the autocontinuity from above of  and  fn → f, we know that for any given ¿0 and m¿m0 , there exists an n1 ¿n0 such that (N 0 −1=2m (fn ))¡ (N 0 −1=m (f)) +  for all n¿n1 . So we have M ¡ S[ 0 ; (N 0 −1=2m (fn ))] 6 S[ 0 ; (N 0 −1=m (f)) + ]:

(3.3)

Letting   0 in (3.3), we obtain (3.1). Letting m → ∞ in (3.1) and noting that (N 0 −1=m (f))  (N H0 (f)), we get S[ 0 ; (N H0 (f))] ¿M . This completes the proof.

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J.-X. Fang / Fuzzy Sets and Systems 124 (2001) 117–123

Lemma 4. Let (X; A; ) be a fuzzy measure space and  be autocontinuous from above. Let {fn } ⊂ M+  and f ∈ M+ be almost everywhere 7nite. If fn → f on A and limn→∞ S[ n ; (N n (fn ))] = 0 for any { n } ⊂ (0; ∞) with n ∞ or n  0; then  lim

n→∞

Lemma 5. Let ∞ be the identity of S and f ∈ M+ . Then  f d6 ∨ (N (f))6 ∨ (N H(f)) X

for any ∈ (0; ∞):

 A

fn d6

A

f d:

Proof. For any ∈ (0; ∞), by (1.1) we have

Proof. Without loss  of generality, we can assume A = X and X f d = c¡∞. Suppose that the conclusion is not true. Then we can take an "¿0 such that limn→∞ A fn d¿c + ". Thus there is a subsequence {fnk } of {fn } such that  f d = sup ¿0 S[ ; (N (fnk ))] ¿ c + ". Hence n k X there exists a sequence { nk } of positive numbers such that S[ nk ; (N nk (fnk ))] ¿ c + ":

 X

f d = sup S[ ; (N (f))] ∈(0; )

∨ sup S[ ; (N (f))] ∈[ ;∞)

6 sup S[ ; ∞] ∈(0; )

∨ sup S[∞; (N (f))]

(3.4)

By Lemma 1, we may as well assume that { nk } is monotone and nk → 0 ∈ [0; ∞]. We consider the following three cases: Case 1: 0 ∈ (0; ∞). Suppose that for all ∈ (0; 0 ), (N (f)) = ∞. Then we have S[ ; ∞] = S[ ; (N (f))]6c. Letting 0 we obtain S[ 0 ; ∞]6c. On the other hand, by (3.4) we have S[ nk ; ∞] ¿ c + ". Letting k → ∞, we obtain S[ 0 ; ∞]¿c + ". This leads to a contradiction. Suppose that there exists a ∈ (0; 0 ) such that  (N (f))¡∞. Noting that fnk → f and (3.4), it is easy to see that three conditions of Lemma 3 are all satis=ed. Hence, by (1.1) and Lemma 3, we obtain

∈[ ;∞)

∈(0; )

which is a contradiction. Case 2: 0 = 0. In this case, it is certain that nk  0. Thus, by the hypothesis and (3.4) we get 0 = lim S[ nk ; (N nk (fnk ))]¿c + ": k→∞

This is a contradiction. Case 3: 0 = ∞. In this case, we have nk ∞. Similarly, from (3.4) it follows that 0¿c + ", which is a contradiction. This completes the proof.

∈[ ;∞)

6 ∨ (N (f))6 ∨ (N H(f)):

4. Convergence theorems of (G) fuzzy integrals We modify Theorem A in the following form. Theorem 1. Let (X; A; ) be a fuzzy measure space.  Then whenever A ∈ A and fn → f on A;  lim

n→∞

c¿S[ 0 ; (N H0 (f))]¿c + ";

∨ sup (N (f))

= sup

 A

fn d =

A

f d;

if and only if  is autocontinuous and limn→∞ S[ n ; (N n (fn ))] = 0 for any { n } ⊂ (0; ∞) with n ∞ or n  0; where {fn } ⊂ M+ and f ∈ M+ is almost everywhere 7nite. Proof. Necessity: For the proof of autocontinuity of , we see Theorem 7:1 in [4].  Now we assume that fn → 0 on X . By the hypothesis, it should hold that   lim fn d = 0 d = 0: n→∞

X

X

J.-X. Fang / Fuzzy Sets and Systems 124 (2001) 117–123

Noting that for any { n } ⊂ (0; ∞) with n ∞ or n  0,  fn d¿S[ n ; (N n (fn ))];

Consequently, by (4.1), (4.2) and (4.5), we obtain  fn d ¿ S[ − ”0 ; (N −”0 (fn ))] X

X

and so we have limn→∞ S[ n ; (N n (fn ))] = 0. Su;ciency: Without loss of generality, we can assume A = X . According to the hypotheses of suMciency, we know that the conditions of Lemma 4 are all satis=ed. Hence, by Lemma 4 we have   lim fn d6 f d:

n→∞

X

X

Thus, it remains only to show that   lim fn d¿ f d:

n→∞

X

X

We may as well assume that X f d = c¿0. So, by (1.1), for any M ∈ (0; c), there is some ¿0 such that S[ ; (N (f))]¿M . From here we know that (N (f))¿0. Thus, by the property of generalized triangular norm S, for any given ”¿0 there exists an ”0 ∈ (0; ) such that S[ ; (N (f))] ¡ S[ − ”0 ; (N (f))] + ”:

(4.1)

Similarly, there exists a "0 ∈ (0; (N (f))) such that S[ − ”0 ; (N (f)) − "0 ] ¿ S[ − ”0 ; (N (f))] − ”:

(4.2)

Moreover, it is easy to see that N (f)\ N”H0 (|fn − f|) ⊂ N −”0 (fn ). Hence, we have (N (f)\N”H0 (|fn − f|))6(N −”0 (fn )):

¿ S[ − ”0 ; (N (f)) − "0 ] ¿ S[ − ”0 ; (N (f))] − ” ¿ S[ ; (N (f))] − 2” ¿ M − 2” for all n¿n0 .  This shows that limn→∞ X fn d¿M . By the arbitrariness of M , we get   fn d¿c = f d: lim

n→∞



(4.3)

121

X

X

This completes the proof. Theorem 2. Let (X; A; ) be a fuzzy measure space,  be autocontinuous and A ∈ A; and let {fn } ⊂ M+ ; f ∈ M+ is almost everywhere 7nite. If  fn → f on A and limn→∞ S[1=n; ∞] = 0; then   fn d = f d: lim n→∞

A

A

Proof. Without loss of generality, we can assume A=X. Assume that for all ∈ (0; ∞), (N (f)) = ∞. Then  f d = sup S[ ; (N (f))] = sup S[ ; ∞] = ∞: ¿0

X

¿0

Since fn → f and  is autocontinuous from below, we have

On the other hand, it is easy to see that N +1 (f)\  N1H (|fn − f|) ⊂ N (fn ). Since, fn → f and  is autocontinuous from below, we have

(N (f)\N”H0 (|fn − f|)) → (N (f)):

(N +1 (f))6 lim (N (fn ));



(4.4)

It follows from (4.3) and (4.4) that (N (f))6 lim (N −”0 (fn )); n→∞

and so for the above "0 , there exists an n0 such that (N −”0 (fn )) ¿ (N (f)) − "0 for all n¿n0 :

(4.5)

n→∞

and so limn→∞ (N (fn )) = ∞. Thus,  fn d¿ lim S[ ; (N (fn ))] = S[ ; ∞]: lim

n→∞

X

n→∞

By the arbitrariness of , we obtain   fn d = ∞ = f d: lim

n→∞

X

X

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J.-X. Fang / Fuzzy Sets and Systems 124 (2001) 117–123

Assume that there exists some ∈ (0; ∞) such that (N (f))¡∞. Then we can prove that for any { n } ⊂ (0; ∞) with n ∞ or n  0, limn→∞ S[ n ; (N n (fn ))] = 0. In fact, if n ∞, then S[ n ; (N n (fn ))]6S[∞; (N n (fn ))]:

(4.6)

Given k, there is an n0 such that for all n¿n0 , we have n ¿k + 1. Thus, N n (fn ) ⊂ N n −1 (f) ∪ N1H (|fn − f|) ⊂ Nk (f) ∪ N1H (|fn − f|): 

Since fn → f and  is autocontinuous from above, we have lim (N n (fn ))6(Nk (f)):

n→∞

∞

Noting that k=1 Nk (f) = {x ∈ X : f(x) = ∞} and f is almost everywhere =nite on X , we have (Nk (f))  0, and so limn→∞ (N n (fn )) = 0. Since S[1=n; ∞] → 0, we have S[∞; (N n (fn ))] → 0 by Lemma 2. Thus, from (4.6) it follows that limn→∞ S[ n ; (N n (fn ))] = 0. If n  0, then S[ n ; (N n (fn ))] ⊂ S[ n ; ∞]. Since S[1=n; ∞] → 0, we have S[ n ; ∞] → 0 by Lemma 2, and so limn→∞ S[ n ; (N n (fn ))] = 0. Consequently, by Theorem 1, we have  lim

n→∞

For (N ) fuzzy integrals (i.e., when S[x; y] = xy), we have the following result. Corollary 2. Let (X; A; ) be a fuzzy measure space.  If (X )¡∞; then whenever A ∈ A and {fn } → f on A;   lim (N ) fn d = (N ) f d; n→∞

A

A

if and only if  is null-additive and limn→∞ n · (N n (fn )) = 0 for any { n } ⊂ (0; ∞) with n ∞; where {fn } ⊂ M+ and f ∈ M+ is almost everywhere 7nite. Proof. The necessity is evident, we have only to prove the suMciency. By Theorems 5:11 and 5:12 in [2], we know that when (X )¡∞, the autocontinuity and nulladditivity are equivalent. Besides, for any n ⊂ (0; ∞), when n  0, S[ n ; (N n (fn ))]6 n · (X ) → 0; when n ∞, by the hypothesis of suMciency, S[ n ; (N n (fn ))] → 0. Consequently, the conclusion follows by Theorem 1. We modify Theorem B in the following form.

 X

fn d =

X

f d:

For (S) fuzzy integrals (i.e., when S[x; y] = min{x; y}) we have the following result. Corollary 1. (Wang and Klir [2, Theorem 7:9]) Let (X; A; ) be a fuzzy measure space. Then whenever  A ∈ A and {fn } → f on A;  lim (S)

n→∞

Proof. Necessity: It follows directly from Theorem 1. Su;ciency: Since S[x; y] = min{x; y}, S[1=n; ∞] → 0, and so the conclusion follows by Theorem 2.

 A

fn d = (S)

A

f d;

if and only if  is autocontinuous. Where {fn } ⊂ M+ ; f ∈ M+ is almost everywhere 7nite.

Theorem 3. Let (X; A; ) be a fuzzy measure space. If  is autocontinuous from above; then whenever  A ∈ A and {fn } → f on A;  |fn − f| d = 0; lim

n→∞

if and only if for any { n } ⊂ (0; ∞) with n ∞ or n  0; limn→∞ S[ n ; (N n (fn ))] = 0; where {fn } ⊂ M+ and f ∈ M+ is almost everywhere 7nite. Proof. Necessity: It is similar to the proof of Theorem 1.  Su;ciency: Suppose that fn → f. Put ’n = |fn −f|  and ’ ≡ 0. Then ’n ; ’ ∈ M+ and ’n → ’. By using

J.-X. Fang / Fuzzy Sets and Systems 124 (2001) 117–123

Lemma 4 to {’n } and ’, we get   lim |fn − f| d6 0 d = 0; n→∞

A

i.e., limn→∞



A

|f − f| d = 0. A n

Theorem 4. Let {fn } ⊂ M+ and f ∈ M+ . If ∞  is the identity of S; then fn → f implies that  if S[ ; ]¿0 limn→∞ |fn − f| d = 0: Conversely;  for any ; ¿0; then limn→∞ |fn − f| d = 0  implies that fn → f. 

123

S. Hence, from Theorem 4 the following corollary directly follows. Corollary 3 (Wang and Klir [2, Theorem 7:4]). For  the (S) fuzzy integrals; fn → f if and only if  lim (S) |fn − f| d = 0: n→∞

X

Corollary 4. For (N ) fuzzy integrals; if  lim (N ) |fn − f| d = 0:

n→∞

X

Proof. If fn → f, then for any given ”¿0 and ∈ (0; ”), there exists n0 such that

then fn → f

(N H(|fn − f|)) ¡ ”

Proof. For the (N ) fuzzy integral, since the correspondent generalized triangular norm is S[x; y] = xy, S[ ; ]¿0 for any ; ¿0. Thus, by Theorem 4, we know that limn→∞ (N ) X |fn −f| d = 0 implies that  fn → f.

whenever n¿n0 :

Since ∞ is the identity of S, by using Lemma 5, we have  |fn − f| d6 ∨ (N H(|fn − f|)) ¡ ”



X

whenever n¿n0 :

Acknowledgements 

This shows that limn→∞ X |fn − f| d = 0.  Conversely, if fn → f does not hold, then there exist ”0 ¿0, "0 ¿0 and a subsequence of natural numbers {ni } such that (N”H0 (|fni −f|)) ¿ "0 ;

i = 1; 2; : : : :

Thus, by (1.1), we have  |fni − f| d ¿ S[”0 ; (N”H0 (|fni − f|))] X

¿ S[”0 ; "0 ] ¿ 0; i = 1; 2; : : : :  This shows that X |fn − f| d 9 0. Obviously, when S[x; y] = min{x; y}, for any ; ¿0 we have S[ ; ]¿0 and ∞ is the identity of

The author is grateful to the referees for their critical comments and helpful suggestions. References [1] M. Sugeno, Theory of fuzzy integrals and its applications, Ph.D. Dissertation, Tokyo Institute of Technology, 1974. [2] Wang Zhen-Yuan, G.J. Klir, Fuzzy Measure Theory, Plenum Press, New York, 1992. [3] Wu Cong-Xin, Wang Shu-Li, Ma Ming, Generalized fuzzy integrals: Part 1. Fundamental concepts, Fuzzy Sets and Systems 57 (1993) 219–226. [4] Wu Cong-Xin, Ma Ming, Song Shi-Ji, Zhang Shao-Tai, Generalized fuzzy integrals: Part 3. Convergent theorems, Fuzzy Sets and Systems 70 (1995) 75–87. [5] Zhao Ru-Huai, (N ) fuzzy integral, J. Math. Res. Exp. 2 (1981) 55–72.