# Generalized Caratheodory Extension Theorem on Fuzzy Measure Space

Authors: Mehmet Şahin; Necati Olgun; F. Talay Akyıldız; Ali Karakuş
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

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.


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## Body

## 1. Introduction

Recently studies including the fuzzy convergence [1], fuzzy soft multiset theory [2], lattices of fuzzy objects [3], on fuzzy soft sets [4], fuzzy sets, fuzzy S-open and S-closed mappings [5], the intuitionistic fuzzy normed space of coefficients [6], set-valued fixed point theorem for generalized contractive mapping on fuzzy metric spaces [7], the centre of the space of Banach lattice-valued continuous functions on the generalized Alexandroff duplicate [8], (L,M)-fuzzy σ-algebras [9], fuzzy number-valued fuzzy measure and fuzzy number-valued fuzzy measure space [10–12], construction of a lattice on the completion space of an algebra and an isomorphism to its Caratheodory extension [13], fuzzy sets [14, 15], generalized σ-algebras and generalized fuzzy measures [16], generalized fuzzy sets [17–19], common fixed points theorems for commutating mappings in fuzzy metric spaces [20], and fuzzy measure theory [21] 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 [13]. In this paper, first we give new definition for lattice-valued-fuzzy measure on [-∞,∞], which is more general than that of [24]. 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 [25], Sahin used the definitions given in [19] and generalized Caratheodory extension theorem for fuzzy sets. In [24], 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, latticeσ-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 L≅L*. We write x≤y if x∧y=x or, equivalently, if x∧y=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 [28]).
If a latticeL satisfies the following conditions, then it is called a lattice σ-algebra.(i)
For allf∈L, fc∈L.(ii)
Iffn∈L for n=1,2,3,…, then ⋁n=1∞fn∈L.It is denoted σ(L) as the lattice σ-algebra generated by L.Definition 2.2 (see [29]).
A lattice-valued setμE is called lattice-valued m*-measurable if for every μA≤μX,
(2.1)m*(μA)=m*(μA∧μE)+m*(μA∧μEC).
This is equivalent to requiring only m*(μA)≥m*(μA∧μE)+m*(μA∧μEC), 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 [29]).
LetμE1 and μE2 be measureable lattice-valued sets. Then,
(2.2)m*(μE1∧μE1C)=0,m*(μE1∨μE2)=m*(μE1)+m*(μE2∧μE1C).


## 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.
Ifm:σ(L)→R∪{∞} satisfies the following properties, then m is called a lattice measure on the lattice σ-algebra σ(L).(i)
m(∅)=L0.(ii)
For allf,g∈σ(L) such that m(f),m(g)≥L0:f≤g⇒m(f)≤m(g).(iii)
For allf,g∈σ(L):m(f∨g)+m(f∧g)=m(f)+m(g).(iv)
fn⊂σ(L),  n∈N such that f1≤f2≤⋯≤fn≤⋯, then m(⋁n=1∞fn)=limn→∞m(fn).Definition 3.2.
Letm1 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), E∈σ(L) is well defined and countable additive σ(L).Definition 3.3.
If a familyσ(L) of membership functions on X satisfies the following conditions, then it is called a lattice fuzzy σ-algebra.(i)
For allα∈L, α∈σ(L), (α constant).(ii)
For allμ∈σ(L), μC=1-μ∈σ(L).(iii)
If(μn)∈σ(L), sup(μn)∈σ(L) for all n∈N.Definition 3.4.
Ifm:σ(L)→R∪{∞} satisfies the following properties, then m is called a lattice-valued fuzzy measure.(i)
m(∅)=L0.(ii)
For allμ1, μ2∈σ(L) such that m(μ1),m(μ2)≥L0:μ1≤μ2⇒m(μ1)≤m(μ2).(iii)
For allμ1, μ2∈σ(L):m(μ1∨μ2)+m(μ1∧μ2)=m(μ1)+m(μ2).(iv)
(μn)∈σ(L),  n∈N such that μ1≤μ2≤⋯≤μn≤⋯; sup(μn)=μ⇒m(μn)=limn→∞m(μn).Definition 3.5.
With a lattice-valued fuzzy outer measurem* having the following properties, we mean an extended lattice-valued set function defined on LX:(i)
m*(∅)=L0,(ii)
m*(μ1)≤m*(μ2)forμ1≤μ2,(iii)
m*(⋁n=1∞μEn)≤(⋁n=1∞m*(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.
IfX 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.
LetF be a class of fuzzy sublattice sets of X containing L0 such that for every μA≤μX, 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 μA∈F. Then, m* is defined on LX by
(3.2)m*(μA)=inf{ψ(μBn)n=1∞:μBn∈F,μA≤μBn},
and m* is a lattice fuzzy outer measure.Proof.
(i)m*(∅)=L0 is obvious.(ii) IfμA1≤μA2 and μA2≤(μBn)n=1∞, then μA1≤(μBn)n=1∞. This means that m*(μA1)≤m(μA2).(iii) LetμEn≤μX for each natural number n. Then, m*(μEn)=∞ for some n. m*(⋁n=1∞μEn)≤(⋁n=1∞m*(μEn)).
The following theorem is an extension of the above proposition.Theorem 3.8.
The classB 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).
Letm be a lattice valued fuzzy measure on a σ-algebra (L)≤LX. Suppose for μE≤μX,         m*(μE)=inf{m(⋁n=1∞μEn):μEn∈σ(L),μE≤⋁n=1∞μEn}.
Then, the following properties are hold.(i)
m* is a lattice-valued fuzzy outer measure.(ii)
μE∈σ(L) implies  m(μE)=m*(μE).(iii)
μE∈σ(L) implies μE is m* lattice fuzzy measurable.(iv)
The restrictionm- 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).(v)
Ifm 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 Proposition3.7.
(ii) Sincem* 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 [29]. Since μE=μE∧(⋁n=1∞μEn)=⋁n=1∞(μE∧μEn) and by the monotonicity and σ-additivity of m, we have m(μE)≤⋁n=1∞m(μE∧μEn)≤⋁n=1∞m(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) LetμE∈σ(L). In order to prove μE is lattice fuzzy measurable, it suffices to show that
(3.5)m*(μA)≥m*(μA∧μE)+m*(μA∧μEc),forμA≤μE.
For givenε>0, there exists μAn∈σ(L), 1≤n<∞ such that
(3.6)⋁n=1∞m(μAn)≤m*(μA)+ε,μA≤⋁n=1∞(μAn).
Now,
(3.7)μA∧μE≤⋁n=1∞(μAn∧μE),μA∧μEc≤⋁n=1∞(μAn∧μEc).
Therefore,
(3.8)m*(μA∧μE)≤⋁n=1∞m(μAn∧μE),m*(μA∧μEc)≤⋁n=1∞m(μAn∧μEc).
From inequalities (3.6) and (3.8), the inequality (3.5) follows.
(iv) Letm- 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,μB∈σ(L-), then μA∨μB∈σ(L-). It also implies that
(3.9)m*(μE)=m*(μE∧μB)+m*(μE∧μBc).
If we write μE∧μA instead of μE in (3.9),
(3.10)m*(μE∧μA)=m*(μE∧μA∧μB)+m*(μE∧μA∧μBc)
is obtained. Now, if we write μE∧μAc  instead of μE in  (3.9),
(3.11)m*(μE∧μAc)=m*(μE∧μAc∧μB)+m*(μE∧μAc∧μBc)
is obtained. If we aggregate with (3.10) and (3.11); we have
(3.12)m*(μE)=m*(μE∧μA∧μB)+m*(μE∧μA∧μBc)+m*(μE∧μAc∧μB)+m*(μE∧μAc∧μBc).
If we write μE∧(μA∨μB) instead of μE in (3.12), then we get
(3.13)m*(μE∧(μA∨μB))=m*(μE∧(μA∨μB)∧μA∧μB)+m*(μE∧(μA∨μB)∧(μAc∧μB))+m*(μE∧(μA∨μB)∧μA∧μBc)+m*(μE∧(μA∨μB)∧μAc∧μBc)=m*(μE∧μA∧μB)+m*(μE∧μAc∧μB)+m*(μE∧μA∧μBc)+m*(L0)=m*(μE∧μA∧μB)+m*(μE∧μAc∧μB)+m*(μE∧μA∧μBc).
From (3.12) and (3.13), we obtain
(3.14)m*(μE)=m*(μE∧(μA∨μB))+m*(μE∧(μA∨μB)c).
Step  2. If μA∈σ(L-), then μAC∈σ(L-). If we write μAc instead of  μA in the equality
(3.15)m*(μE)=m*(μE∧μA)+m*(μE∧μAc),
we have
(3.16)m*(μE)=m*(μE∧μAc)+m*(μE∧(μAc)c);(μAc)c=μA=m*(μE∧μAc)+m*(μE∧μA)=m*(μE).
Therefore, it follows that μAC∈σ(L-). Therefore, we showed that σ(L-) is the algebra of lattice sets.
Step  3. Let μA,μB∈σ(L) and μA∧μB=∅, From (3.13), we have
(3.17)m*(μE∧(μA∨μB))=m*(μE∧μAc∧μB)+m*(μE∧μA∧μBc)=m*(μE∧μB)+m*(μE∧μA).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*(μE∧μBn).
LetμA=⋁n=1∞μAn and μAn∈σ(L). Then, μA=⋁n=1∞μBn, μBn=(μAn∧(⋁x=1n-1μAx)c), and μBi∧μBj=∅ for i≠j. Therefore, we obtain following inequality:
(3.19)m*(μE)≥m*(μE∧(⋁n=1∞μBn))+m*(μE∧(⋁n=1∞μBn)c).
Hence, m* is a lattice σ-semiadditive.
Sinceσ(L-) is an algebra, ⋁n=1kμBn∈σ(L-) for all n∈N. 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=1∞μBn)c≤μE∧(⋁n=1∞μBn)c and monotonicity of lattice-valued fuzzy measure and (3.20), we have
(3.21)m*(μE)≥⋁j=1nm*(μE∧μBj)+m*(μE∧μAc).
Then, taking the limit of both sides, we get
(3.22)m*(μE)≥⋁j=1∞m*(μE∧μBj)+m*(μE∧μAc).
Using the semiadditivity, we have,
(3.23)m*(μE∧μA)=m*(⋁j=1∞(μE∧μBj))=m*(μE∧(⋁j=1∞μBj))≤m*(μE∧μBj).
From (3.22), we have
(3.24)m*(μE)≥m*(μE∧μA)+m*(μE∧μAc).
Hence, μA∈σ(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.
LetμE=⋁j=1∞μAj. From (3.22), we have
(3.25)m*(⋁j=1∞μAj)≥⋁j=1∞m*(μAj).
Step  6. We have σ(L-)⊃σ(L).
LetμA∈σ(L) and μE≤μA. Then, we must show the following inequality:
(3.26)m*(μE)≥m*(μE∧μA)+m*(μE∧μAc).
IfμE∈σ(L), then μE∧μA and μE∧μAc are different and both of them belong to σ(L), (3.26) is obvious and since m*=m, hence additive.
WithμE≤μX and given ε>0, σ(L), there is μEj which contains  σ(L) such that we have
(3.27)m*(μE)+ε>⋁j=1∞m(μEj).
Now, from the equality(3.28)μEj=(μEj∧μA)∨(μEj∧μAc)
and from the Definition 2.1 and Theorem 2.3, we have the following equality:
(3.29)m(μEj)=m(μEj∧μA)+m(μEj∧μAc).
Therefore, we obtain the following:(3.30)μE∧μA≤⋁j=1∞(μEj∧μA),μE∧μAc≤⋁j=1∞(μEj∧μAc).
Using the monotonicity and semiadditivity, we obtain
(3.31)m*(μE∧μA)≤⋁j=1∞m(μEj∧μA),m*(μE∧μAc)≤⋁j=1∞m(μEj∧μAc).
Using the sum of the inequalities (3.31),
(3.32)m*(μE∧μA)+m*(μEj∧μAc)≤⋁j=1∞m*(μEj)<m*(μE)+ε
is obtained. For arbitrary ε>0, (3.26) is proven. Therefore, (iv) it is obtained as required.
(v) Letσ(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 μE∈σ(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=1∞μEn,μEn∈σ(L),n≠k,μEn∧μEk=∅,m(μEn)<∞;1≤n<∞.
If μA∈σ(L-), then we have
(3.35)m¯(μB)=⋁n=1∞m¯(μA∧μEn),m1(μA)=⋁n=1∞m1(μA∧μEn).
To prove the inequality (3.33), it suffices to show that
(3.36)m1(μA)=m¯(μA),μA∈σ(L-),m¯(μA)<∞.
LetμA∈σ(L¯), m¯(μA)<∞, and ε>0 arbitrary. Then, we have
(3.37)μA≤⋁n=1∞μEn,forμEn∈σ(L),1≤n<∞,(3.38)m¯(⋁n=1∞μEn)≤⋁n=1∞m(μEn)<m¯(μA)+ε.
Since m1(μA)≤m1(⋁n=1∞μEn)≤⋁n=1∞m1(μEn)=⋁n=1∞m(μEn) and from (3.38), we get
(3.39)m1(μA)≤m¯(μA).
Also, from (3.38), we can write μF=∨n=1∞μEn∈σ(L¯) for the sets μEn. Therefore, μF is m* lattice fuzzy measurable. From the inequality μA≤μF and (3.38),
(3.40)m¯(μF)=m¯(μA)+m¯(μF-μA),m¯(μF-μA)=m¯(μF)-m¯(μA)<ε
are obtained.
From the equalitiesm(μE)=m¯(μE) and m1(μF)=m¯(μF) for all μA∈σ(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 fuzzyσ-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 assumeσ(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 μA∈σ(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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