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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 260457, 11 pages
Generalized Caratheodory Extension Theorem on Fuzzy Measure Space
Department of Mathematics, Faculty of Arts and Sciences, University of Gaziantep, 27310 Gaziantep, Turkey
Received 23 April 2012; Revised 28 June 2012; Accepted 10 October 2012
Academic Editor: Alberto D'Onofrio
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.
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 , ()-fuzzy -algebras , 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 , fuzzy sets [14, 15], generalized -algebras and generalized fuzzy measures , generalized fuzzy sets [17–19], 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 -algebra, and a finitely additive measure on the algebra to a countable additive measure on the -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 . 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 -algebra, are given. In Section 3, definitions for lattice-valued fuzzy -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.
In this section, we shall briefly review the well-known facts about lattice theory [26, 27], purpose an extension lattice, and investigate its properties. or simply under closed operations , is called a lattice. For two lattices and , a bijection from to , which preserves lattice operations is called a lattice isomorphism, or simply an isomorphism. If there is an isomorphism from to , then is called a lattice isomorphic with , and we write . We write if or, equivalently, if . is called complete, if any subset of includes the supremum and infimum , with respect to the above order. A complete lattice includes the maximum and minimum elements, which are denoted and .
Throughout this paper, will be denoted the entire set and is a lattice of any subset sets of .
Definition 2.1 (see ). If a lattice satisfies the following conditions, then it is called a lattice -algebra.(i)For all , .(ii)If for , then .It is denoted as the lattice -algebra generated by .
Definition 2.2 (see ). A lattice-valued set is called lattice-valued -measurable if for every ,
This is equivalent to requiring only , since the converse inequality is obvious from the subadditive property of .
Also, is a class of all lattice-valued measurable sets.
Theorem 2.3 (see ). Let and be measureable lattice-valued sets. Then,
3. Main Results
Throughout this paper, we will consider lattices as complete lattices, will denote space, and μ is a membership function of any fuzzy set .
Definition 3.1. If satisfies the following properties, then is called a lattice measure on the lattice -algebra .(i). (ii)For all such that .(iii)For all .(iv), such that , then .
Definition 3.2. Let and be lattice measures defined on the same lattice -algebra . If one of them is finite, the set function , is well defined and countable additive .
Definition 3.3. If a family of membership functions on satisfies the following conditions, then it is called a lattice fuzzy -algebra.(i)For all , , ( constant). (ii)For all , . (iii) If , for all .
Definition 3.4. If satisfies the following properties, then is called a lattice-valued fuzzy measure.(i).(ii)For all , such that . (iii)For all , . (iv), such that ; .
Definition 3.5. With a lattice-valued fuzzy outer measure having the following properties, we mean an extended lattice-valued set function defined on :(i),(ii), (iii).
Example 3.6. Suppose
is infimum of sets of lattice family, and is supremum of sets of lattice family.
If has at least two member, then is a lattice-valued fuzzy outer measure which is not lattice-valued fuzzy measure on .
Proposition 3.7. Let be a class of fuzzy sublattice sets of containing such that for every , there exists a sequence from such that . Let be an extended lattice-valued function on such that and for . Then, is defined on by and is a lattice fuzzy outer measure.
Proof. (i) is obvious. (ii) If and , then . This means that . (iii) Let for each natural number . Then, for some . .
The following theorem is an extension of the above proposition.
Theorem 3.8. The class of lattice-valued fuzzy measurable sets is a -algebra. Also, the restriction of to 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 be a lattice valued fuzzy measure on a -algebra . Suppose for , .
Then, the following properties are hold.(i) is a lattice-valued fuzzy outer measure.(ii) implies .(iii) implies is lattice fuzzy measurable.(iv)The restriction of to the -lattice-valued fuzzy measurable sets in an extension of to a lattice-valued fuzzy measure on a fuzzy -algebra containing .(v)If is lattice-valued fuzzy -finite, then is the only lattice fuzzy measure (on the smallest fuzzy - algebra containing that is an extension of ).
Proof. (i) It follows from Proposition 3.7.
(ii) Since is a lattice-valued fuzzy outer measure, we have For given , there exists such that . Since and by the monotonicity and -additivity of , we have . Since is arbitrary, we conclude that
From (3.3) and (3.4), is obtained.
(iii) Let . In order to prove is lattice fuzzy measurable, it suffices to show that
For given , there exists , such that Now, Therefore, From inequalities (3.6) and (3.8), the inequality (3.5) follows.
(iv) Let be the restriction of to the lattice-valued measurable sets, when we write . Now, we must show that is a lattice fuzzy -algebra containing and is a lattice-valued fuzzy measure on . We show it step by stepin the following.
Step 1. If , then . It also implies that If we write instead of in (3.9), is obtained. Now, if we write instead of in (3.9), is obtained. If we aggregate with (3.10) and (3.11); we have If we write instead of in (3.12), then we get From (3.12) and (3.13), we obtain
Step 2. If , then . If we write instead of in the equality we have Therefore, it follows that . Therefore, we showed that is the algebra of lattice sets.
Step 3. Let and , From (3.13), we have Step 4. is a lattice -algebra.
From the previous step, we have for every family of (for each disjoint lattice sets) ,
Let and . Then, , , and for . Therefore, we obtain following inequality: Hence, is a lattice -semiadditive.
Since is an algebra, for all . The following inequality is satisfied for all : From the inequality and monotonicity of lattice-valued fuzzy measure and (3.20), we have Then, taking the limit of both sides, we get Using the semiadditivity, we have, From (3.22), we have Hence, . This shows that is a lattice fuzzy -algebra.
Step 5. is a lattice fuzzy measure, where we only need to show lattice is -additive.
Let . From (3.22), we have
Step 6. We have .
Let and . Then, we must show the following inequality:
If , then and are different and both of them belong to , (3.26) is obvious and since , hence additive.
With and given , , there is which contains such that we have
Now, from the equality and from the Definition 2.1 and Theorem 2.3, we have the following equality:
Therefore, we obtain the following: Using the monotonicity and semiadditivity, we obtain Using the sum of the inequalities (3.31), is obtained. For arbitrary , (3.26) is proven. Therefore, (iv) it is obtained as required.
(v) Let be the smallest -algebra which contain the and let be a lattice fuzzy measure on . Then, for all . We must show that Since is a finite -lattice fuzzy measure, we can write If , then we have To prove the inequality (3.33), it suffices to show that
Let , , and arbitrary. Then, we have Since and from (3.38), we get Also, from (3.38), we can write for the sets . Therefore, is lattice fuzzy measurable. From the inequality and (3.38), are obtained.
From the equalities and for all , we can write Therefore, from (3.41), 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 -finiteness assumption is essential in generalized Caratheodory extension theorem for the uniqueness of the extension of on the smallest fuzzy -algebra containing .
In this example, let we assume is the smallest fuzzy -algebra containing . And let be the smallest fuzzy -algebra containing . Otherwise, let be lattice family such that and For if , and if .
After all these, solution of application is clearly in the generalized Caratheodory extension theorem at property (v).
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