Almost everywhere convergence of random set sequence on non-additive measure spaces ∗ Guiling Li1
Jun Li1 † ‡ Masami Yasuda2
1. Department of Applied Mathematics, Southeast University Nanjing 210096, China 2. Department of Mathematics & Informatics, Faculty of Science, Chiba University Chiba 263-8522, Japan
November 4, 2004 ABSTRACT: In this paper, we investigate the conver- is its lower limit. A subset A is said to be the set limit of the gence and the pseudo-convergence of sequence of set-valued sequence {An }, denoted by limn→∞ An = A, if mappings. Egoroff type theorem for random set sequence on lim sup An = lim inf An = A monotone non-additive measure spaces is presented. n→∞ n→∞ Keyword: Set-valued mapping; Random set; Non-additive measure; Egoroff’s theorem where d(x, An ) = infy∈A d(x, y). Definition 2 ([4]) A set function µ : A → [0, +∞] is called monotone non-additive measure, if it satisfies the following properties: / = 0; (1) µ(0) (2) µ(A) ≤ µ(B) whenever A ⊂ B and A, B ∈ A (monotonicity). When µ is a monotone non-additive measure, the triple (Ω, A , µ) is called monotone non-additive measure space.
1 Introduction The convergences of random set sequence on measure spaces were studied by Zhang ([10]), and some important convergence theorems in random set theory were obtained. Liu ([6]) discussed the convergence of sequence of random set (it is called measurable set-valued function in [6]) on fuzzy measure spaces and some results, such as Egoroff’s theorem, Lebesgue’s theorem and Riesz’s theorem in fuzzy measure theory ([9]) have been adapted to set-valued case. In this paper, we discuss the convergence and the pseudoconvergence of sequence of random set sequence on monotone non-additive measure spaces. Egoroff type theorem for random set sequence with respect to monotone non-additive measure is shown. It is a generalization of the related results obtained by authors [3, 4, 5].
A set function µ : A → [0, +∞] is said to be continuous from below, if limn→∞ µ(An ) = µ(A) whenever An % A ([7]); strongly order continuous ([4]), if limn→+∞ µ(An ) = 0 whenever {An }n ⊂ F , An & B and µ(B) = 0; has property (S) (resp. property (PS)) ([8]), if for any {An }n with limn→∞ µ(An ) = 0 (resp. limn→∞ µ(A \ An ) = µ(A)), there exists a subsequence {Ani }i of {An }n such that µ(lim sup Ani ) = 0 (resp. µ(A \ lim sup Ani ) = µ(A)). Definition 3 ([10]) Let (Ω, A ) be a measurable space, and f a set-valued mapping from Ω to closed subsets of Rm . If for every closed subset F of Rm ,
2 Preliminaries Let Ω be a non-empty set, A be a σ-algebra of subsets of Ω, and let Rm be m-dimensional Euclidean space, and d a Euclidean metric on Rm . All concepts and signs not defined in this paper may be found in [1, 2, 10]
/ ∈ A, f −1 (F) = {ω ∈ Ω : f (ω) ∩ F 6= 0} then f is called random set (with respect to A ).
Let µ[Ω] denote the class of all random set defined on Ω Definition 1 Let {An } be a sequence of closed subsets of Rm . (with respect to A ), and let fn (n ∈ N), f ∈ µ(Ω), E ∈ A . We We say that the subset say that (1) { fn } converges to f almost everywhere on E, and delim sup An = {x ∈ Rm : lim inf d(x, An ) = 0} n→∞ a.e. n→∞ noted by fn −→ f on E, if there exists N ∈ E ∩ A , such that µ(N) = 0 and for every ω ∈ E \ N, limn→∞ fn (ω) = f (ω) (in is the upper limit of the sequence {An } and that the subset the sense of Definition 1); u (2) { fn } converges uniformly to f on E , denoted by fn −→ lim inf An = {x ∈ Rm : lim d(x, An ) = 0} n→∞ n→∞ f on E, if for any ε > 0 and any compact subset K of Rm , there ∗ Project (10371017) supported by the National Natural Science Foundaexists some positive integer N(ε,K) , such that tion of China. † Corresponding ‡ The
author. Email address:
[email protected] author was supported by the China Scholarship Council.
/ = 0/ E(4−1 εn (K)) , {ω ∈ A : [( f n \ ε f ) ∪ ( f \ ε f n )](ω) ∩ K 6= 0} 1
whenever n ≥ N(ε,K) ; (3) { fn } converges almost uniformly to f on E, denoted by a.u fn −→ f on E, if there exists a sequence {Em } of measurable u sets of E ∩ A such that limn→∞ µ(Em ) = 0, and fn −→ f on E \ Em (m = 1, 2 . . .); (4) { fn } converges to f pseudo−almost everywhere on E, p.a.e. denote by fn −→ f on E, if there exists N ∈ E ∩ A , such that µ(N) = µ(E) and for every ω ∈ E \ N, limn→∞ fn (ω) = f (ω); (5) { fn } pseudo−converges almost uniformly to f on E, p.a.u denoted by fn −→ f on E, if there exists a sequence {Em } of measurable sets of E ∩ A such that limn→∞ µ(E \ Em ) = µ(E), u and fn −→ f on E \ Em (m = 1, 2 . . .) (cf. [6], [9] and [10]).
3
Egoroff’s theorem for random set sequence
By using the strong order continuity of µ, we have (l)
lim µ(X − Em ) = µ(X − E) = 0 (l ≥ 1).
m→∞
(l)
(l)
Thus, there exists a subsequence {X \ Eml } of {X \ Em } satisfying 1 (l) µ(X \ Eml ) ≤ (∀l ≥ 1), l and hence (l) lim µ(X \ Eml ) = 0. l→∞
By applying the property (S) of µ to the sequence {X \ Eml l },
then there exists a subsequence {X \ Emli li } of {X \ Eml l } such that ¶ µ (l )
µ lim sup(X \ Emlii ) = 0 i
and l1 < l2 < . . .. On the other hand, ∞ [
In this section, we show Egoroff type theorem for random set sequence on monotone non-additive measure space.
(l )
(l )
(X \ Emljj ) & lim sup(X \ Emlii ) ( j → ∞), i
j=k
Theorem 1 (Egoroff’s theorem) Let µ be a monotone non- therefore, by using the strong order continuity of µ, we have ! Ã additive measure on (Ω, A ) and fn (n ∈ N), f ∈ µ(Ω) and E ∈ ∞ [ (l j ) A. (X \ Eml j ) = 0. lim µ k→∞ (1) If µ is strongly order continuous and has property (S), j=k then on E ∞ a.e. a.u \ (l ) fn −→ f =⇒ fn −→ f . Put Ek = Emljj , then limn→∞ µ(X \ Ek ) = 0. j=k (2) If µ is continuous from below and has property (PS), then Now we prove that fn converges to f on Ek uniformly for on E p.a.e. p.a.u any fixed k = 1, 2, . . .. In fact, for any ε > 0 and any compact fn −→ f =⇒ fn −→ f . subset K of Rm , there exists some positive integer l0(ε,K) such S Proof: (1) Let 0 < εl ↓ 0, and Rm = ∞ l=1 Ul , where Ul is that εl0 < ε and K ⊂ Ul0 . Therefore we have a bounded open subset of Rm , and its closure Ul ⊂ Ul+1 (l = ∞ \ (l0 ) 1, 2, . . .). / {ω ∈ X : [( fn \ ε f ) ∪ ( f \ ε fn )](ω) ∩ K = 0} ⊂ E ⊂ E m k l a.e. 0 Since fn −→ f , there exists E ∈ A , such that µ(X − E) = 0, n=ml0 and fn converges to f everywhere on E. that is, For each l > 0, denote (l) Em
=
Ek (4−1 εn (K))
∞ © [
ª ω ∈ X : [( fn \ ε f ) ∪ ( f \ ε fn )](ω) ∩Ul = 0/ ,
/ = {ω ∈ X | [( fn \ ε f ) ∪ ( f \ ε fn )](ω) ∩ K 6= 0}
n=m
= 0/
(l)
then Em is increasing in n for each fixed l, and we get S∞ (l) (l) ∞ m=1 Em = E(l = 1, 2, . . .). In fact, for any ω ∈ ∪m=1 Em , there exists m0 , such that (l)
ω ∈ Em0 =
∞ © [
ª
ω ∈ X : [( fn \ ε f ) ∪ ( f \ ε fn )](ω) ∩Ul = 0/ ,
n=m0
that is, h i [ \ ( fn \ εl f ) ( f \ εl fn ) (ω) Ul = 0/ whenever n ≥ m0 . Then for any ε > 0 and any compact subset Kof Rm , there exists some positive integer l0 (ε, K), such / So that εl0 < ε, K ⊂ Ul0 , and [( fn \ ε f ) ∪ ( f \ ε fn )](ω) ∩ K = 0. we have
(l) ∪∞ m=1 Em
= E(l = 1, 2, . . .), and for each fixed l (l)
X − Em & X −
∞ [ m=1
a.u
whenever n ≥ N(ε,K) = ml0 . This shows fn −→ f . (2) Let εl > 0, εl ↓ 0 and Rm =
∞ [
Ul , where Ul is a bounded
l=1
open subset of Rm , and its closure Ul ⊂ Ul+1 (l = 1, 2, . . .). p.a.e. Since fn −→ f on A, there exists E ∈ A ∩ A , such that µ(E) = µ(A) and fn converges to f everywhere on E. For each l > 0, letting (l)
Em =
∞ \
/ {ω ∈ A : [( fn \ ε f ) ∪ ( f \ ε fn )](ω) ∩ K = 0},
n=m (l)
∞ [
(l)
then E1 ⊂ E2 ⊂ . . ., and
(l)
Em = E (l = 1, 2, . . .). By us-
m=1
ing the continuity from below of µ, we have (l)
Em = X − E.
(l)
lim µ(Em ) = µ(E) = µ(A).
m→∞
(l)
(l)
Thus there exists a subsequence {Eml }l of {Em ; l, m ≥ 1} satisfying: (i) if µ(A) < ∞, then 1 (l) µ(A) − µ(Eml ) < , ∀ l ≥ 1. l
[5] J. Li, Y. Ouyang, M. Yasuda, Pseudoconvergence of measurable functions on Sugeno fuzzy measure spaces, Proceedings of 7th Joint Conference on Information Science, North Corolina, USA, September 26-30, pp. 56-59.
(2) if µ(A) = ∞, then (l)
µ(Eml ) > l, ∀ l ≥ 1. Therefore, we have
[6] Y. Liu, Convergence of measurable set-valued function sequence on fuzzy measure space, Fuzzy Sets and Systems 112 (2000) 241−249.
(l)
lim µ(Eml ) = µ(A).
l→∞
(l)
By applying the property (PS) of µ to the sequence {Eml }, (l) (l ) then there exists a subsequence {Emlii } of {Eml }, such that l1 < l2 < . . . and ! Ã ∞ \ ∞ [ (li )
µ
Emli
= µ(A).
k=1 i=k
k→∞
Put Fk = A \
∞ \ (li )
Emli
i=k
∞ \ (li )
Emli (k = 1, 2 . . .), then lim µ(A \ Fk ) = µ(A).
k→∞
Now we prove that fn converges to f on A\Fk uniformly for any fixed k = 1, 2, . . . In fact, for any ε > 0, and any compact subset K of Rm , there exists some positive integer l0(ε,K) , such that εl0 < ε and K ⊂ Ul0 , So we have ∞ \
[8] Q. Sun, Property (s) of fuzzy measure and Riesz’s theorem, Fuzzy Sets and Systems 62 (1994) 117−119.
[10] W. Zhang, Set-Valued Measure Theory and Random Sets, Xian Jiaotong University, 1987 (in Chinese).
= µ(A).
i=k
A \ Fk ⊂
[7] E. Pap, Null-additive Set Functions, Kluwer, Dordrecht, 1995.
[9] Z. Wang and G. J. Klir, Fuzzy Measure Theory, Plenum, New York, 1992.
It follows from the continuity from below of µ that ! Ã lim µ
[4] J. Li, M. Yasuda, Egoroff’s theorems on monotone non-additive measure space, Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems 12 (2004) 61-68.
/ {ω ∈ A : [( fn \ ε f ) ∪ ( f \ ε fn )](ω) ∩ K = 0}.
j=ml0
That is A \ Fk (4−1 εn (K)) / = {ω ∈ A : [( fn \ ε f ) ∪ ( f \ ε fn )](ω) ∩ K 6= 0} = 0/ p.a.u
whenever n ≥ N(ε,K) = ml0 . Therefore, we have fn −→ f . 2 REFERENCES [1] R. J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 121 (1965) 1−12. [2] P. Diamond, P. Kloeden, Metric Space of Fuzzy Sets – Theory and Applicatuions, World Scientific, Singapore, 1994. [3] J. Li, On Egoroff’s theorems on fuzzy measure space, Fuzzy Sets and Systems, 135 (2003) 367−375.
