AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 260457 10.1155/2012/260457 260457 Research Article Generalized Caratheodory Extension Theorem on Fuzzy Measure Space Şahin Mehmet Olgun Necati Akyıldız F. Talay Karakuş Ali D'Onofrio Alberto Department of Mathematics Faculty of Arts and Sciences University of Gaziantep 27310 Gaziantep Turkey gantep.edu.tr 2012 8 11 2012 2012 23 04 2012 28 06 2012 10 10 2012 2012 Copyright © 2012 Mehmet Şahin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Lattice-valued fuzzy measures are lattice-valued set functions which assign the bottom element of the lattice to the empty set and the top element of the lattice to the entire universe, satisfying the additive properties and the property of monotonicity. In this paper, we use the lattice-valued fuzzy measures and outer measure definitions and generalize the Caratheodory extension theorem for lattice-valued fuzzy measures.

1. Introduction

Recently studies including the fuzzy convergence , fuzzy soft multiset theory , lattices of fuzzy objects , on fuzzy soft sets , fuzzy sets, fuzzy S-open and S-closed mappings , the intuitionistic fuzzy normed space of coefficients , set-valued fixed point theorem for generalized contractive mapping on fuzzy metric spaces , the centre of the space of Banach lattice-valued continuous functions on the generalized Alexandroff duplicate , (L,M)-fuzzy σ-algebras , fuzzy number-valued fuzzy measure and fuzzy number-valued fuzzy measure space , construction of a lattice on the completion space of an algebra and an isomorphism to its Caratheodory extension , fuzzy sets [14, 15], generalized σ-algebras and generalized fuzzy measures , generalized fuzzy sets , common fixed points theorems for commutating mappings in fuzzy metric spaces , and fuzzy measure theory  have been investigated.

The well-known Caratheodory extension theorem in classical measure theory is very important [22, 23]. In a graduate course in real analysis, students learn the Caratheodory extension theorem, which shows how to extended an algebra to a σ-algebra, and a finitely additive measure on the algebra to a countable additive measure on the σ-algebra . In this paper, first we give new definition for lattice-valued-fuzzy measure on [-,], which is more general than that of . Using this new definition, we provide new proof of Caratheodory extension theorem for lattice valued-fuzzy measure. In related literature, not many studies have been explored including Caratheodory extension theorem on lattice-valued fuzzy measure. In , Sahin used the definitions given in  and generalized Caratheodory extension theorem for fuzzy sets. In , lattice-valued fuzzy measure and fuzzy integral were studied on [0,]. However, no study has been done related to Caratheodory extension theorem for lattice-valued fuzzy measure. This provides the motivation for present paper where we provide the proof of generalized Caratheodory extension theorem for lattice-valued fuzzy measure space.

The outline of the paper is as follows. In the next section, basic definitions of lattice theory, lattice σ-algebra, are given. In Section 3, definitions for lattice-valued fuzzy σ-algebra, and lattice-valued fuzzy outer measure are given, and some necessary theorems for our main theorem (generalized caratheodory extension theorem) related to lattice-valued fuzzy outer measure and main theorem of this paper are given.

2. Preliminaries

In this section, we shall briefly review the well-known facts about lattice theory [26, 27], purpose an extension lattice, and investigate its properties. (L,,) or simply L under closed operations , is called a lattice. For two lattices L and L*, a bijection from L to L*, which preserves lattice operations is called a lattice isomorphism, or simply an isomorphism. If there is an isomorphism from L to L*, then L is called a lattice isomorphic with L*, and we write LL*. We write xy if xy=x or, equivalently, if xy=y. L is called complete, if any subset A of L includes the supremum A and infimum A, with respect to the above order. A complete lattice L includes the maximum and minimum elements, which are denoted L1 and L0.

Throughout this paper, X will be denoted the entire set and L is a lattice of any subset sets of X.

Definition 2.1 (see [<xref ref-type="bibr" rid="B18">28</xref>]).

If a lattice L satisfies the following conditions, then it is called a lattice σ-algebra.

For all fL, fcL.

If fnL for n=1,2,3,, then n=1fnL.

It is denoted σ(L) as the lattice σ-algebra generated by L.

Definition 2.2 (see [<xref ref-type="bibr" rid="B26">29</xref>]).

A lattice-valued set μE is called lattice-valued m*-measurable if for every μAμX, (2.1)m*(μA)=m*(μAμE)+m*(μAμEC). This is equivalent to requiring only m*(μA)m*(μAμE)+m*(μAμEC), since the converse inequality is obvious from the subadditive property of m*.

Also, M={μE:μEism*-measurable} is a class of all lattice-valued measurable sets.

Theorem 2.3 (see [<xref ref-type="bibr" rid="B26">29</xref>]).

Let μE1 and μE2 be measureable lattice-valued sets. Then, (2.2)m*(μE1μE1C)=0,m*(μE1μE2)=m*(μE1)+m*(μE2μE1C).

3. Main Results

Throughout this paper, we will consider lattices as complete lattices, X will denote space, and μ is a membership function of any fuzzy set X.

Definition 3.1.

If m:σ(L)R{} satisfies the following properties, then m is called a lattice measure on the lattice σ-algebra σ(L).

m()=L0.

For all f,gσ(L) such that m(f),m(g)L0:fgm(f)m(g).

For all f,gσ(L):m(fg)+m(fg)=m(f)+m(g).

fnσ(L), nN such that f1f2fn, then m(n=1fn)=limn  m(fn).

Definition 3.2.

Let m1 and m2 be lattice measures defined on the same lattice σ-algebra σ(L). If one of them is finite, the set function m(E)=m1(E)-m2(E), Eσ(L) is well defined and countable additive σ(L).

Definition 3.3.

If a family σ(L) of membership functions on X satisfies the following conditions, then it is called a lattice fuzzy σ-algebra.

For all αL, ασ(L), (α constant).

For all   μσ(L), μC=1-μσ(L).

If (μn)σ(L), sup(μn)σ(L) for all nN.

Definition 3.4.

If m:σ(L)R{} satisfies the following properties, then m is called a lattice-valued fuzzy measure.

m()=L0.

For all   μ1, μ2σ(L) such that m(μ1),m(μ2)L0:μ1μ2m(μ1)m(μ2).

For all μ1, μ2σ(L):m(μ1μ2)+m(μ1μ2)=m(μ1)+m(μ2).

(μn)σ(L), nN such that μ1μ2μn; sup(μn)=μ      m(μn)=limnm(μn).

Definition 3.5.

With a lattice-valued fuzzy outer measure m* having the following properties, we mean an extended lattice-valued set function defined on LX:

m*()=L0,

m*(μ1)m*(μ2)  for  μ1  μ2,

m*(n=1μEn)(n=1m*(μEn)).

Example 3.6.

Suppose (3.1)m*={L0,μE=,L1,μE.L0 is infimum of sets of lattice family, and L1 is supremum of sets of lattice family.

If X has at least two member, then m* is a lattice-valued fuzzy outer measure which is not lattice-valued fuzzy measure on LX.

Proposition 3.7.

Let F be a class of fuzzy sublattice sets of X containing L0 such that for every μAμX, there exists a sequence (μBn)n=1 from F such that μA(μBn)n=1. Let ψ be an extended lattice-valued function on F such that ψ()=L0 and ψ(μA)L0 for μAF. Then, m* is defined on LX by (3.2)m*(μA)=inf{ψ(μBn)n=1:μBnF,μAμBn}, and m* is a lattice fuzzy outer measure.

Proof.

(i) m*()=L0 is obvious.

(ii) If μA1μA2 and μA2(μBn)n=1, then μA1(μBn)n=1. This means that m*(μA1)m(μA2).

(iii) Let μEnμX for each natural number n. Then, m*(μEn)= for some n. m*(n=1μEn)(n=1m*(μEn)).

The following theorem is an extension of the above proposition.

Theorem 3.8.

The class B of m* lattice-valued fuzzy measurable sets is a σ-algebra. Also, m- the restriction m* of to B is a lattice valued fuzzy measure.

Proof.

It follows from extension of the proposition.

Now, we shall generalize the well-known Caratheodory extension theorem in classical measure theory for lattice-valued fuzzy measure.

Theorem 3.9 (Generalized Caratheodory Extension Theorem).

Let m be a lattice valued fuzzy measure on a σ-algebra (L)LX. Suppose for μEμX, m*(μE)= inf {m(n=1μEn):μEnσ(L),  μEn=1μEn}.

Then, the following properties are hold.

m* is a lattice-valued fuzzy outer measure.

μEσ(L) implies m(μE)=m*(μE).

μEσ(L) implies μE is m* lattice fuzzy measurable.

The restriction m- of m* to the m*-lattice-valued fuzzy measurable sets in an extension of m to a lattice-valued fuzzy measure on a fuzzy σ-algebra containing (L).

If m is lattice-valued fuzzy σ-finite, then m- is the only lattice fuzzy measure (on the smallest fuzzy σ- algebra containing σ(L) that is an extension of m).

Proof.

(i) It follows from Proposition 3.7.

(ii) Since m* is a lattice-valued fuzzy outer measure, we have (3.3)m*(μE)m(μE). For given ε>0, there exists (μEn;n=1,2,) such that n=1(m(μEn))m*(μE)+ε . Since μE=μE(n=1μEn)=n=1(μEμEn) and by the monotonicity and σ-additivity of m, we have m(μE)n=1m(μEμEn)n=1m(μEn)m*(μE)+ε. Since ε>0 is arbitrary, we conclude that (3.4)m(μE)m*(μE).

From (3.3) and (3.4), m(μE)=m*(μE) is obtained.

(iii) Let μEσ(L). In order to prove μE is lattice fuzzy measurable, it suffices to show that (3.5)m*(μA)m*(μAμE)+m*(μAμEc),for μAμE.

For given ε>0, there exists μAnσ(L), 1n< such that (3.6)n=1m(μAn)m*(μA)+ε,μAn=1(μAn). Now, (3.7)μAμEn=1(μAnμE),μAμEcn=1(μAnμEc). Therefore, (3.8)m*(μAμE)n=1m(μAnμE),m*(μAμEc)n=1m(μAnμEc). From inequalities (3.6) and (3.8), the inequality (3.5) follows.

(iv) Let m- be the restriction of m* to the m* lattice-valued measurable sets, when we write m-=m*/σ(L-). Now, we must show that σ(L-) is a lattice fuzzy σ-algebra containing σ(L) and m- is a lattice-valued fuzzy measure on σ(L). We show it step by stepin the following.

Step  1. If μA,μBσ(L-), then μAμBσ(L-). It also implies that (3.9)m*(μE)=m*(μEμB)+m*(μEμBc). If we write μEμA instead of μE in (3.9), (3.10)m*(μEμA)=m*(μEμAμB)+m*(μEμAμBc) is obtained. Now, if we write μEμAc instead of μE in (3.9), (3.11)m*(μEμAc)=m*(μEμAcμB)+m*(μEμAcμBc) is obtained. If we aggregate with (3.10) and (3.11); we have (3.12)m*(μE)=m*(μEμAμB)+m*(μEμAμBc)+m*(μEμAcμB)+m*(μEμAcμBc). If we write μE(μAμB) instead of μE in (3.12), then we get (3.13)  m*(μE(μAμB))=m*(μE(μAμB)μAμB)+m*(μE(μAμB)(μAcμB))+m*(μE(μAμB)μAμBc)+m*(μE(μAμB)μAcμBc)=m*(μEμAμB)+m*(μEμAcμB)+m*(μEμAμBc)+m*(L0)=m*(μEμAμB)+m*(μEμAcμB)+m*(μEμAμBc). From (3.12) and (3.13), we obtain (3.14)m*(μE)=m*(μE(μAμB))+m*(μE(μAμB)c).

Step  2. If μAσ(L-), then μACσ(L-). If we write μAc instead of μA in the equality (3.15)m*(μE)=m*(μEμA)+m*(μEμAc), we have (3.16)m*(μE)=m*(μEμAc)+m*(μE(μAc)c);(μAc)c=μA=m*(μEμAc)+m*(μEμA)=m*(μE). Therefore, it follows that μACσ(L-). Therefore, we showed that σ(L-) is the algebra of lattice sets.

Step  3. Let μA,μBσ(L) and μAμB=, From (3.13), we have (3.17)m*(μE(μAμB))=m*(μEμAcμB)+m*(μEμAμBc)  =m*(μEμB)+m*(μEμA).Step  4. σ(L-) is a lattice σ-algebra.

From the previous step, we have for every family of (for each disjoint lattice sets) (μBn),n=1,2,, (3.18)m*(μE(n=1kμBn))=n=1km*(μEμBn).

Let μA=n=1μAn and μAnσ(L). Then, μA=n=1μBn, μBn=(μAn(x=1n-1μAx)c), and μBiμBj= for ij. Therefore, we obtain following inequality: (3.19)m*(μE)m*(μE(n=1μBn))+m*(μE(n=1μBn)c). Hence, m* is a lattice σ-semiadditive.

Since σ(L-) is an algebra, n=1kμBnσ(L-) for all nN. The following inequality is satisfied for all n: (3.20)m*(μE)m*(μE(n=1kμBn))+m*(μE(n=1kμBn)c). From the inequality μE(n=1μBn)cμE(n=1μBn)c and monotonicity of lattice-valued fuzzy measure and (3.20), we have (3.21)m*(μE)j=1nm*(μEμBj)+m*(μEμAc). Then, taking the limit of both sides, we get (3.22)m*(μE)j=1m*(μEμBj)+m*(μEμAc). Using the semiadditivity, we have, (3.23)m*(μEμA)=m*(j=1(μEμBj))=m*(μE(j=1μBj))m*(μEμBj). From (3.22), we have (3.24)m*(μE)m*(μEμA)+m*(μEμAc). Hence, μAσ(L-). This shows that σ(L-) is a lattice fuzzy σ-algebra.

Step  5. m-=m*/σ(L-) is a lattice fuzzy measure, where we only need to show lattice is σ-additive.

Let μE=j=1μAj. From (3.22), we have (3.25)m*(j=1μAj)j=1m*(μAj).

Step  6. We have σ(L-)σ(L).

Let μAσ(L) and μEμA. Then, we must show the following inequality: (3.26)m*(μE)m*(μEμA)+m*(μEμAc).

If μEσ(L), then μEμA and μEμAc are different and both of them belong to σ(L), (3.26) is obvious and since m*=m, hence additive.

With  μEμX and given ε>0, σ(L), there is μEj which contains  σ(L) such that we have (3.27)m*(μE)+ε>j=1m(μEj).

Now, from the equality (3.28)μEj=(μEjμA)(μEjμAc) and from the Definition 2.1 and Theorem 2.3, we have the following equality: (3.29)m(μEj)=m(μEjμA)+m(μEjμAc).

Therefore, we obtain the following: (3.30)μEμAj=1(μEjμA),μEμAcj=1(μEjμAc). Using the monotonicity and semiadditivity, we obtain (3.31)m*(μEμA)j=1m(μEjμA),m*(μEμAc)j=1m(μEjμAc). Using the sum of the inequalities (3.31), (3.32)m*(μEμA)+m*(μEjμAc)j=1m*(μEj)<m*(μE)+ε is obtained. For arbitrary ε>0, (3.26) is proven. Therefore, (iv) it is obtained as required.

(v) Let σ(L-) be the smallest σ-algebra which contain the σ(L) and let m1 be a lattice fuzzy measure on σ(L-). Then, m1(μE)=m(μE) for all μEσ(L). We must show that (3.33)m1(μA)=m¯(μA). Since m is a finite σ-lattice fuzzy measure, we can write (3.34)X=n=1μEn,μEnσ(L),nk,μEnμEk=,m(μEn)<;1n<. If μAσ(L-), then we have (3.35)m¯(μB)=n=1m¯(μAμEn),m1(μA)=n=1m1(μAμEn). To prove the inequality (3.33), it suffices to show that (3.36)m1(μA)=m¯(μA),μAσ(L-),m¯(μA)<.

Let  μAσ(L¯), m¯(μA)<, and ε>0 arbitrary. Then, we have (3.37)μAn=1μEn,for  μEnσ(L),1n<,(3.38)m¯(n=1μEn)n=1m(μEn)<m¯(μA)+ε. Since m1(μA)m1(n=1μEn)n=1m1(μEn)=n=1m(μEn) and from (3.38), we get (3.39)m1(μA)m¯(μA). Also, from (3.38), we can write μF=n=1μEnσ(L¯) for the sets μEn. Therefore, μF is m* lattice fuzzy measurable. From the inequality μAμF and (3.38), (3.40)m¯(μF)=m¯(μA)+m¯(μF-μA),m¯(μF-μA)=m¯(μF)-m¯(μA)<ε are obtained.

From the equalities m(μE)=m¯(μE) and m1(μF)=m¯(μF) for all μAσ(L), we can write (3.41)m(μA)m(μF)=m1(μF)=m1(μA)+m1(μF-μA)m1(μA)+m¯(μF-μA). Therefore, from (3.41), (3.42)m¯(μA)m1(μA) is obtained.

Finally from the inequalities (3.41) and (3.39), hence the proof is completed.

An Application of Generalized Caratheodory Extension Theorem

An application of generalized Caratheodory extension theorem is in the following. This application is essentially related to option (v)th of the generalized Caratheodory extension theorem.

Example 3.10.

Show that the lattice-valued fuzzy σ-finiteness assumption is essential in generalized Caratheodory extension theorem for the uniqueness of the extension of m on the smallest fuzzy σ-algebra containing σ(L).

In this example, let we assume σ(L-) is the smallest fuzzy σ-algebra containing σ(L). And let σ(L-) be the smallest fuzzy σ-algebra containing σ(L). Otherwise, let L be lattice family such that L=(L0,L1] and (3.43)σ(L)={i=1(L0i,L1i]:(L0i,L1i](L0,L1]}. For μAσ(L),m(μA)= if μA, and m(μA)=L0 if μA=.

After all these, solution of application is clearly in the generalized Caratheodory extension theorem at property (v).

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