Generalized Caratheodory Extension Theorem on Fuzzy Measure Space

and Applied Analysis 3 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 . i m ∅ L0. ii For all f, g ∈ σ L such that m f , m g ≥ L0 : f ≤ g ⇒ m f ≤ m g . iii For all f, 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 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 , E ∈ σ 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. i For all α ∈ L, α ∈ σ L , α constant . ii For all μ ∈ σ L , μ 1 − μ ∈ σ L . iii If μn ∈ σ L , sup μn ∈ σ L for all n ∈ N. Definition 3.4. If m : σ L → R ∪ {∞} satisfies the following properties, then m is called a lattice-valued fuzzy measure. i m ∅ L0. ii For all μ1, μ2 ∈ σ L such that m μ1 , m μ2 ≥ L0 : μ1 ≤ μ2 ⇒ m μ1 ≤ m μ2 . iii For all μ1, 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 L : i m∗ ∅ L0, ii m∗ μ1 ≤ m∗ μ2 for μ1 ≤ μ2, iii m∗ ∨∞ n 1μEn ≤ ∨∞ n 1m ∗ μEn . Example 3.6. Suppose m∗ { L0, μE ∅, L1, μE / ∅. 3.1 L0 is infimum of sets of lattice family, and L1 is supremum of sets of lattice family. 4 Abstract and Applied Analysis If X has at least two member, thenm∗ is a lattice-valued fuzzy outer measure which is not lattice-valued fuzzy measure on L . Proposition 3.7. Let F 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 L by m∗ ( μA ) inf { ψ ( μBn )∞ n 1 : μBn ∈ F, μA ≤ μBn } , 3.2 and m∗ is a lattice fuzzy outer measure. Proof. i m∗ ∅ L0 is obvious. ii If μA1 ≤ μA2 and μA2 ≤ μBn n 1, then μA1 ≤ μBn n 1. This means that m∗ μA1 ≤ m μA2 . iii Let μEn ≤ μX 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 restrictionm∗ 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 ≤ L . Suppose for μE ≤ μX , m∗ μE inf {m ∨∞ n 1μEn : μEn ∈ σ L , μE ≤ ∨∞ n 1μEn}. Then, the following properties are hold. i m∗ is a lattice-valued fuzzy outer measure. ii μE ∈ σ L impliesm μE m∗ μE . iii μE ∈ σ L implies μE ism∗ lattice fuzzy measurable. iv The restrictionm ofm∗ to them∗-lattice-valued fuzzy measurable sets in an extension ofm to a lattice-valued fuzzy measure on a fuzzy σ-algebra containing L . v Ifm is lattice-valued fuzzy σ-finite, thenm is the only lattice fuzzy measure (on the smallest fuzzy σalgebra 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 m∗ ( μE ) ≤ mμE ) . 3.3 For given ε > 0, there exists μEn ;n 1, 2, . . . such that ∨∞ n 1 m μEn ≤ m∗ μE ε 29 . Since μE μE ∧ ∨∞ n 1μEn ∨∞ n 1 μE ∧μEn and by the monotonicity and σ-additivity ofm, we have Abstract and Applied Analysis 5 m μE ≤ ∨∞ n 1m μE ∧ μEn ≤ ∨∞ n 1m μEn ≤ m∗ μE ε. Since ε > 0 is arbitrary, we conclude thatand Applied Analysis 5 m μE ≤ ∨∞ n 1m μE ∧ μEn ≤ ∨∞ n 1m μEn ≤ m∗ μE ε. Since ε > 0 is arbitrary, we conclude that m ( μE ) ≤ m∗μE ) . 3.4 From 3.3 and 3.4 , m μE m∗ μE is obtained. iii Let μE ∈ σ L . In order to prove μE is lattice fuzzy measurable, it suffices to show that m∗ ( μA ) ≥ m∗μA ∧ μE ) m∗ ( μA ∧ μcE ) , for μA ≤ μE. 3.5 For given ε > 0, there exists μAn ∈ σ L , 1 ≤ n < ∞ such that ∞ ∨ n 1 m ( μAn ) ≤ m∗μA ) ε, μA ≤ ∞ ∨