" Almost everywhere convergence of random set sequence on non-additive measure spaces "

(1)

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-gence and the pseudo-converconver-gence of sequence of set-valued mappings. Egoroff type theorem for random set sequence on monotone non-additive measure spaces is presented.

Keyword: Set-valued mapping; Random set; Non-additive

measure; Egoroff’s theorem

1 Introduction

The convergences of random set sequence on measure spaces were studied by Zhang ([10]), and some important con-vergence 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 pseudo-convergence of sequence of random set sequence on mono-tone 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].

2 Preliminaries

Let Ω be a non-empty set,

A

be a σ-algebra of subsets ofΩ, 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]

Definition 1 Let {An} be a sequence of closed subsets of Rm. We say that the subset

lim sup n→∞

An= {x ∈ Rm: lim inf

n→∞ d(x, An) = 0} is the upper limit of the sequence {An} and that the subset

lim inf

n→∞ An= {x ∈ R m_{: lim}

n→∞d(x, An) = 0}

∗_{Project (10371017) supported by the National Natural Science} Founda-tion of China.

†_{Corresponding author. Email address: [email protected]} ‡_{The author was supported by the China Scholarship Council.}

is its lower limit. A subset A is said to be the set limit of the sequence {An}, denoted by limn→∞An= A, if

lim sup n→∞

An= lim inf n→∞ An= A where d(x, An) = infy∈Ad(x, y).

Definition 2 ([4]) A set function µ :

A

→ [0, +∞] is called

monotone non-additive measure, if it satisfies the following

properties: (1) µ(/0) = 0;

(2) µ(A) ≤ µ(B) whenever A ⊂ B and A, B ∈

A

(mono-tonicity).

When µ is a monotone non-additive measure, the triple

(Ω,

A

, µ) is called monotone non-additive measure space.

A set function µ :

A

→ [0, +∞] is said to be

continu-ous 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

prop-erty (S) (resp. propprop-erty (PS)) ([8]), if for any {An}n with limn→∞µ(An) = 0 (resp. limn→∞µ(A \ An) = µ(A)), there ex-ists a subsequence {Ani}iof {An}nsuch 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 fromΩto closed subsets of Rm. If for every closed subset F of Rm,

f−1(F) = {ω∈Ω: f (ω) ∩ F 6=/0} ∈

A

, then f is called random set (with respect to

A

).

Let µ[Ω] denote the class of all random set defined onΩ

(with respect to

A

), and let fn(n ∈ N), f ∈ µ(Ω), E ∈

A. We

say that

(1) { fn} converges to f almost everywhere on E, and de-noted by fn

a.e.

−→ f on E, if there exists N ∈ E ∩

A

, such that

µ(N) = 0 and for everyω∈ E \ N, limn→∞fn(ω) = f (ω) (in the sense of Definition 1);

(2) { fn} converges uniformly to f on E , denoted by fn u

−→

f on E, if for anyε> 0 and any compact subset K of Rm, there exists some positive integer N₍_ε_,K), such that

E(4−1_ε_n(K)) , {ω∈ A : [( fn\εf ) ∪ ( f \εfn)](ω) ∩ K 6=/0} =/0 1

(2)

whenever n ≥ N₍_ε_,K);

(3) { fn} converges almost uniformly to f on E, denoted by

fn a.u

−→ f on E, if there exists a sequence {Em} of measurable

sets of E ∩

A

such that limn→∞µ(Em) = 0, and fn u

−→ f on

E \ Em(m = 1, 2 . . .);

(4) { fn} converges to f pseudo−almost everywhere on E, denote by fn

p.a.e.

−→ f on E, if there exists N ∈ E ∩

A

, such that

µ(N) = µ(E) and for everyω∈ E \ N, limn→∞fn(ω) = f (ω); (5) { fn} pseudo−converges almost uniformly to f on E, denoted by fn

p.a.u

−→ f on E, if there exists a sequence {Em} of measurable sets of E ∩A such that limn→∞µ(E \ Em) = µ(E), and fn−→ f on E \ Eu m(m = 1, 2 . . .) (cf. [6], [9] and [10]).

3 Egoroff’s theorem for random set

sequence

In this section, we show Egoroff type theorem for random set sequence on monotone non-additive measure space.

Theorem 1 (Egoroff ’s theorem) Let µ be a monotone

non-additive measure on (Ω,

A

) and fn(n ∈ N), f ∈ µ(Ω) and E ∈

A

.

(1) If µ is strongly order continuous and has property (S),

then on E fn a.e. −→ f =⇒ fn a.u −→ f .

(2) If µ is continuous from below and has property (PS), then

on E fn p.a.e. −→ f =⇒ fn p.a.u −→ f . Proof: (1) Let 0 <εl ↓ 0, and Rm =

S_∞

l=1Ul, where Ul is a bounded open subset of Rm, and its closure Ul⊂ Ul+1(l = 1, 2, . . .).

Since fn a.e.

−→ f , there exists E ∈

A

, such that µ(X − E) = 0, and fnconverges to f everywhere on E.

For each l > 0, denote

Em(l)= ∞ [ n=m © ω∈ X : [( fn\εf ) ∪ ( f \εfn)](ω) ∩Ul=/0 ª ,

then Em(l) is increasing in n for each fixed l, and we get

S_∞

m=1E

(l)

m = E(l = 1, 2, . . .). In fact, for any ω∈ ∪∞_m=1Em(l), there exists m0, such that

ω∈ Em(l)0= ∞ [ n=m0 © ω∈ X : [( fn\εf ) ∪ ( f \εfn)](ω) ∩Ul=/0 ª , that is, h ( fn\εlf ) [ ( f \εlfn) i (ω)\Ul = /0

whenever n ≥ m0. Then for anyε> 0 and any compact subset Kof Rm, there exists some positive integer l0(ε, K), such thatεl0<ε, K ⊂ Ul0, and [( fn\εf ) ∪ ( f \εfn)](ω) ∩ K =/0. So we have ∪∞_m=1Em(l)= E(l = 1, 2, . . .), and for each fixed l

X − Em(l)& X −

∞

[

m=1

Em(l)= X − E.

By using the strong order continuity of µ, we have lim

m→∞µ(X − E

(l)

m ) = µ(X − E) = 0 (l ≥ 1).

Thus, there exists a subsequence {X \ Em(l)l} of {X \ E

(l) m } sat-isfying µ(X \ Em(l)l) ≤ 1 l (∀l ≥ 1), and hence lim l→∞µ(X \ E (l) ml) = 0.

By applying the property (S) of µ to the sequence {X \ E_ml

l},

then there exists a subsequence {X \ Eli

m_li} of {X \ Emll} such that µ µ lim sup i (X \ E(li) m_li) ¶ = 0

and l1< l2< . . .. On the other hand,

∞

[

j=k

(X \ E(lj)

m_{l j}) & lim sup i

(X \ E(li)

m_li) ( j →∞), therefore, by using the strong order continuity of µ, we have

lim k→∞µ Ã ∞ [ j=k (X \ E(lj) m_{l j}) ! = 0. Put Ek= ∞ \ j=k Em(l_{l j}j), then limn→∞µ(X \ Ek) = 0.

Now we prove that fnconverges to f on Ek uniformly for any fixed k = 1, 2, . . .. In fact, for anyε> 0 and any compact subset K of Rm, there exists some positive integer l₀₍_ε_,K)such thatεl0<εand K ⊂ Ul0. Therefore we have

Ek⊂ Em(l0_l0)⊂ ∞ \ n=m_l0 {ω∈ X : [( fn\εf ) ∪ ( f \εfn)](ω) ∩ K =/0} that is, Ek(4−1εn(K)) = {ω∈ X | [( fn\εf ) ∪ ( f \εfn)](ω) ∩ K 6=/0} = /0

whenever n ≥ N₍_ε_,K)= ml0. This shows fn a.u −→ f . (2) Letεl> 0,εl↓ 0 and Rm= ∞ [ l=1 Ul, where Ulis a bounded open subset of Rm_{, and its closure U}

l⊂ Ul+1(l = 1, 2, . . .). Since fn

p.a.e.

−→ 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 Em(l)= ∞ \ n=m {ω∈ A : [( fn\εf ) ∪ ( f \εfn)](ω) ∩ K =/0}, then E₁(l)⊂ E₂(l)⊂ . . ., and ∞ [ m=1 Em(l)= E (l = 1, 2, . . .). By us-ing the continuity from below of µ, we have

lim m→∞µ(E

(l)

(3)

Thus there exists a subsequence {Em(l)l}lof {E

(l)

m ; l, m ≥ 1} sat-isfying:

(i) if µ(A) <∞, then

µ(A) − µ(Em(l)l) < 1 l, ∀ l ≥ 1. (2) if µ(A) =∞, then µ(Em(l)l) > l, ∀ l ≥ 1. Therefore, we have lim l→∞µ(E (l) ml) = µ(A).

By applying the property (PS) of µ to the sequence {Em(l)l},

then there exists a subsequence {E(li)

m_li} of {Em(l)l}, such that l1< l2< . . . and µ Ã ∞ [ k=1 ∞ \ i=k E(li) m_li ! = µ(A).

It follows from the continuity from below of µ that

lim k→∞µ Ã ∞ \ i=k E(li) m_li ! = µ(A). Put Fk= A \ ∞ \ i=k E(li) m_li (k = 1, 2 . . .), then lim k→∞µ(A \ Fk) = µ(A).

Now we prove that fnconverges to f on A \ Fkuniformly for any fixed k = 1, 2, . . . In fact, for anyε> 0, and any compact subset K of Rm_{, there exists some positive integer l}

0(ε,K), such

thatε_l₀<εand K ⊂ U_l₀, So we have

A \ Fk⊂ ∞ \ j=m_l0 {ω∈ A : [( fn\εf ) ∪ ( f \εfn)](ω) ∩ K =/0}. That is A \ Fk(4−1εn(K)) = {ω∈ A : [( fn\εf ) ∪ ( f \εfn)](ω) ∩ K 6=/0} = /0

whenever n ≥ N₍_ε_,K)= ml0. Therefore, we have fn p.a.u

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