Generalized Caratheodory Extension Theorem on Fuzzy Measure Space

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.

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, 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 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 lattice 𝐿 satisfies the following conditions, then it is called a lattice 𝜎-algebra.(i)For all 𝑓∈𝐿, 𝑓^𝑐∈𝐿.(ii)If 𝑓_𝑛∈𝐿 for 𝑛=1,2,3,…, then ⋁^∞_𝑛=1𝑓_𝑛∈𝐿.It is denoted 𝜎(𝐿) as the lattice 𝜎-algebra generated by 𝐿.

Definition 2.2 (see [29]). A lattice-valued set 𝜇_𝐸 is called lattice-valued 𝑚^∗-measurable if for every 𝜇_𝐴≤𝜇_𝑋, 𝑚^∗𝜇_𝐴=𝑚^∗𝜇_𝐴∧𝜇_𝐸+𝑚^∗𝜇_𝐴∧𝜇^𝐶_𝐸.(2.1) This is equivalent to requiring only 𝑚^∗(𝜇_𝐴)≥𝑚^∗(𝜇_𝐴∧𝜇_𝐸)+𝑚^∗(𝜇_𝐴∧𝜇^𝐶_𝐸), since the converse inequality is obvious from the subadditive property of 𝑚^∗.
Also, 𝑀={𝜇_𝐸∶𝜇_𝐸is𝑚^∗-measurable} is a class of all lattice-valued measurable sets.

Theorem 2.3 (see [29]). Let 𝜇_𝐸1 and 𝜇_𝐸2 be measureable lattice-valued sets. Then, 𝑚^∗𝜇_𝐸1∧𝜇^𝐶_𝐸1𝑚=0,_∗𝜇_𝐸1∨𝜇_𝐸2=𝑚^∗𝜇_𝐸1+𝑚^∗𝜇_𝐸2∧𝜇^𝐶_𝐸1.(2.2)

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)𝑓_𝑛⊂𝜎(𝐿), 𝑛∈𝑁 such that 𝑓₁≤𝑓₂≤⋯≤𝑓_𝑛≤⋯, then 𝑚(⋁^∞_𝑛=1𝑓_𝑛)=lim_𝑛→∞𝑚(𝑓_𝑛).

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^𝐶=1−𝜇∈𝜎(𝐿). (iii) If (𝜇_𝑛)∈𝜎(𝐿), sup(𝜇_𝑛)∈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)(𝜇_𝑛)∈𝜎(𝐿), 𝑛∈𝑁 such that 𝜇₁≤𝜇₂≤⋯≤𝜇_𝑛≤⋯; sup(𝜇_𝑛)=𝜇⇒𝑚(𝜇_𝑛)=lim_𝑛→∞𝑚(𝜇_𝑛).

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 𝐿^𝑋:(i)𝑚^∗(∅)=𝐿₀,(ii)𝑚^∗(𝜇₁)≤𝑚^∗(𝜇₂)for𝜇₁≤𝜇₂, (iii)𝑚^∗(⋁^∞_𝑛=1𝜇_𝐸𝑛)≤(⋁^∞_𝑛=1𝑚^∗(𝜇_𝐸𝑛)).

Example 3.6. Suppose 𝑚^∗=𝐿₀,𝜇_𝐸𝐿=∅,₁,𝜇_𝐸≠∅.(3.1)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 𝐿^𝑋.

Proposition 3.7. Let 𝐹 be a class of fuzzy sublattice sets of 𝑋 containing 𝐿₀ such that for every 𝜇_𝐴≤𝜇_𝑋, there exists a sequence (𝜇_𝐵𝑛)^∞_𝑛=1 from 𝐹 such that 𝜇_𝐴≤(𝜇_𝐵𝑛)^∞_𝑛=1. Let 𝜓 be an extended lattice-valued function on 𝐹 such that 𝜓(∅)=𝐿₀ and 𝜓(𝜇_𝐴)≥𝐿₀ for 𝜇_𝐴∈𝐹. Then, 𝑚^∗ is defined on 𝐿^𝑋 by 𝑚^∗𝜇_𝐴=inf𝜓𝜇_𝐵𝑛^∞_𝑛=1∶𝜇_𝐵𝑛∈𝐹,𝜇_𝐴≤𝜇_𝐵𝑛,(3.2) and 𝑚^∗ is a lattice fuzzy outer measure.

Proof. (i) 𝑚^∗(∅)=𝐿₀ is obvious. (ii) If 𝜇_𝐴1≤𝜇_𝐴2 and 𝜇_𝐴2≤(𝜇_𝐵𝑛)^∞_𝑛=1, then 𝜇_𝐴1≤(𝜇_𝐵𝑛)^∞_𝑛=1. This means that 𝑚^∗(𝜇_𝐴1)≤𝑚(𝜇_𝐴2). (iii) Let 𝜇_𝐸𝑛≤𝜇_𝑋 for each natural number 𝑛. Then, 𝑚^∗(𝜇_𝐸𝑛)=∞ for some 𝑛. 𝑚^∗(⋁^∞_𝑛=1𝜇_𝐸𝑛)≤(⋁^∞_𝑛=1𝑚^∗(𝜇_𝐸𝑛)).
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)≤𝐿^𝑋. Suppose for 𝜇_𝐸≤𝜇_𝑋, 𝑚^∗(𝜇_𝐸)=inf{𝑚(⋁^∞_𝑛=1𝜇_𝐸𝑛)∶𝜇_𝐸𝑛∈𝜎(𝐿),𝜇_𝐸≤⋁^∞_𝑛=1𝜇_𝐸𝑛}.
Then, the following properties are hold.(i)𝑚^∗ is a lattice-valued fuzzy outer measure.(ii)𝜇_𝐸∈𝜎(𝐿) implies 𝑚(𝜇_𝐸)=𝑚^∗(𝜇_𝐸).(iii)𝜇_𝐸∈𝜎(𝐿) implies 𝜇_𝐸 is 𝑚^∗ lattice fuzzy measurable.(iv)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).(v)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 𝑚^∗𝜇_𝐸≤𝑚𝜇_𝐸.(3.3) For given 𝜀>0, there exists (𝜇_𝐸𝑛;𝑛=1,2,…) such that ⋁^∞_𝑛=1(𝑚(𝜇_𝐸𝑛))≤𝑚^∗(𝜇_𝐸)+𝜀 [29]. Since 𝜇_𝐸=𝜇_𝐸∧(⋁^∞_𝑛=1𝜇_𝐸𝑛)=⋁^∞_𝑛=1(𝜇_𝐸∧𝜇_𝐸𝑛) and by the monotonicity and 𝜎-additivity of 𝑚, we have 𝑚(𝜇_𝐸)≤⋁^∞_𝑛=1𝑚(𝜇_𝐸∧𝜇_𝐸𝑛)≤⋁^∞_𝑛=1𝑚(𝜇_𝐸𝑛)≤𝑚^∗(𝜇_𝐸)+𝜀. Since 𝜀>0 is arbitrary, we conclude that 𝑚𝜇_𝐸≤𝑚^∗𝜇_𝐸.(3.4)
From (3.3) and (3.4), 𝑚(𝜇_𝐸)=𝑚^∗(𝜇_𝐸) is obtained.
(iii) Let 𝜇_𝐸∈𝜎(𝐿). In order to prove 𝜇_𝐸 is lattice fuzzy measurable, it suffices to show that 𝑚^∗𝜇_𝐴≥𝑚^∗𝜇_𝐴∧𝜇_𝐸+𝑚^∗𝜇_𝐴∧𝜇^𝑐_𝐸,for𝜇_𝐴≤𝜇_𝐸.(3.5)
For given 𝜀>0, there exists 𝜇_𝐴𝑛∈𝜎(𝐿), 1≤𝑛<∞ such that ^∞_𝑛=1𝑚𝜇_𝐴𝑛≤𝑚^∗𝜇_𝐴+𝜀,𝜇_𝐴≤^∞_𝑛=1𝜇_𝐴𝑛.(3.6) Now, 𝜇_𝐴∧𝜇_𝐸≤^∞_𝑛=1𝜇_𝐴𝑛∧𝜇_𝐸,𝜇_𝐴∧𝜇^𝑐_𝐸≤^∞_𝑛=1𝜇_𝐴𝑛∧𝜇^𝑐_𝐸.(3.7) Therefore, 𝑚^∗𝜇_𝐴∧𝜇_𝐸≤^∞_𝑛=1𝑚𝜇_𝐴𝑛∧𝜇_𝐸,𝑚^∗𝜇_𝐴∧𝜇^𝑐_𝐸≤^∞_𝑛=1𝑚𝜇_𝐴𝑛∧𝜇^𝑐_𝐸.(3.8) 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 𝜇_𝐴,𝜇_𝐵∈𝜎(𝐿), then 𝜇_𝐴∨𝜇_𝐵∈𝜎(𝐿). It also implies that 𝑚^∗𝜇_𝐸=𝑚^∗𝜇_𝐸∧𝜇_𝐵+𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐵.(3.9) If we write 𝜇_𝐸∧𝜇_𝐴 instead of 𝜇_𝐸 in (3.9), 𝑚^∗𝜇_𝐸∧𝜇_𝐴=𝑚^∗𝜇_𝐸∧𝜇_𝐴∧𝜇_𝐵+𝑚^∗𝜇_𝐸∧𝜇_𝐴∧𝜇^𝑐_𝐵(3.10) is obtained. Now, if we write 𝜇_𝐸∧𝜇^𝑐_𝐴 instead of 𝜇_𝐸 in (3.9), 𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐴=𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐴∧𝜇_𝐵+𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐴∧𝜇^𝑐_𝐵(3.11) is obtained. If we aggregate with (3.10) and (3.11); we have 𝑚^∗𝜇_𝐸=𝑚^∗𝜇_𝐸∧𝜇_𝐴∧𝜇_𝐵+𝑚^∗𝜇_𝐸∧𝜇_𝐴∧𝜇^𝑐_𝐵+𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐴∧𝜇_𝐵+𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐴∧𝜇^𝑐_𝐵.(3.12) If we write 𝜇_𝐸∧(𝜇_𝐴∨𝜇_𝐵) instead of 𝜇_𝐸 in (3.12), then we get 𝑚^∗𝜇_𝐸∧𝜇_𝐴∨𝜇_𝐵=𝑚^∗𝜇_𝐸∧𝜇_𝐴∨𝜇_𝐵∧𝜇_𝐴∧𝜇_𝐵+𝑚^∗𝜇_𝐸∧𝜇_𝐴∨𝜇_𝐵∧𝜇^𝑐_𝐴∧𝜇_𝐵+𝑚^∗𝜇_𝐸∧𝜇_𝐴∨𝜇_𝐵∧𝜇_𝐴∧𝜇^𝑐_𝐵+𝑚^∗𝜇_𝐸∧𝜇_𝐴∨𝜇_𝐵∧𝜇^𝑐_𝐴∧𝜇^𝑐_𝐵=𝑚^∗𝜇_𝐸∧𝜇_𝐴∧𝜇_𝐵+𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐴∧𝜇_𝐵+𝑚^∗𝜇_𝐸∧𝜇_𝐴∧𝜇^𝑐_𝐵+𝑚^∗𝐿₀=𝑚^∗𝜇_𝐸∧𝜇_𝐴∧𝜇_𝐵+𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐴∧𝜇_𝐵+𝑚^∗𝜇_𝐸∧𝜇_𝐴∧𝜇^𝑐_𝐵.(3.13) From (3.12) and (3.13), we obtain 𝑚^∗𝜇_𝐸=𝑚^∗𝜇_𝐸∧𝜇_𝐴∨𝜇_𝐵+𝑚^∗𝜇_𝐸∧𝜇_𝐴∨𝜇_𝐵^𝑐.(3.14)
Step  2. If 𝜇_𝐴∈𝜎(𝐿), then 𝜇^𝐶_𝐴∈𝜎(𝐿). If we write 𝜇^𝑐_𝐴 instead of 𝜇_𝐴 in the equality 𝑚^∗𝜇_𝐸=𝑚^∗𝜇_𝐸∧𝜇_𝐴+𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐴,(3.15) we have 𝑚^∗𝜇_𝐸=𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐴+𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐴^𝑐;𝜇^𝑐_𝐴^𝑐=𝜇_𝐴=𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐴+𝑚^∗𝜇_𝐸∧𝜇_𝐴=𝑚^∗𝜇_𝐸.(3.16) Therefore, it follows that 𝜇^𝐶_𝐴∈𝜎(𝐿). Therefore, we showed that 𝜎(𝐿) is the algebra of lattice sets.
Step  3. Let 𝜇_𝐴,𝜇_𝐵∈𝜎(𝐿) and 𝜇_𝐴∧𝜇_𝐵=∅, From (3.13), we have 𝑚^∗𝜇_𝐸∧𝜇_𝐴∨𝜇_𝐵=𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐴∧𝜇_𝐵+𝑚^∗𝜇_𝐸∧𝜇_𝐴∧𝜇^𝑐_𝐵=𝑚^∗𝜇_𝐸∧𝜇_𝐵+𝑚^∗𝜇_𝐸∧𝜇_𝐴.(3.17)Step  4. 𝜎(𝐿) is a lattice 𝜎-algebra.
From the previous step, we have for every family of (for each disjoint lattice sets) (𝜇_𝐵𝑛),𝑛=1,2,…, 𝑚^∗𝜇_𝐸∧^𝑘_𝑛=1𝜇_𝐵𝑛=^𝑘_𝑛=1𝑚^∗𝜇_𝐸∧𝜇_𝐵𝑛.(3.18)
Let 𝜇_𝐴=⋁^∞_𝑛=1𝜇_𝐴𝑛 and 𝜇_𝐴𝑛∈𝜎(𝐿). Then, 𝜇_𝐴=⋁^∞_𝑛=1𝜇_𝐵𝑛, 𝜇_𝐵𝑛=(𝜇_𝐴𝑛∧(⋁^𝑛−1_𝑥=1𝜇_𝐴𝑥)^𝑐), and 𝜇_𝐵𝑖∧𝜇_𝐵𝑗=∅ for 𝑖≠𝑗. Therefore, we obtain following inequality: 𝑚^∗𝜇_𝐸≥𝑚^∗𝜇_𝐸∧^∞_𝑛=1𝜇_𝐵𝑛+𝑚^∗𝜇_𝐸∧^∞_𝑛=1𝜇_𝐵𝑛^𝑐.(3.19) Hence, 𝑚^∗ is a lattice 𝜎-semiadditive.
Since 𝜎(𝐿) is an algebra, ⋁^𝑘_𝑛=1𝜇_𝐵𝑛∈𝜎(𝐿) for all 𝑛∈𝑁. The following inequality is satisfied for all 𝑛: 𝑚^∗𝜇_𝐸≥𝑚^∗𝜇_𝐸∧^𝑘_𝑛=1𝜇_𝐵𝑛+𝑚^∗𝜇_𝐸∧^𝑘_𝑛=1𝜇_𝐵𝑛^𝑐.(3.20) From the inequality 𝜇_𝐸∧(⋁^∞_𝑛=1𝜇_𝐵𝑛)^𝑐≤𝜇_𝐸∧(⋁^∞_𝑛=1𝜇_𝐵𝑛)^𝑐 and monotonicity of lattice-valued fuzzy measure and (3.20), we have 𝑚^∗𝜇_𝐸≥^𝑛_𝑗=1𝑚^∗𝜇_𝐸∧𝜇_𝐵𝑗+𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐴.(3.21) Then, taking the limit of both sides, we get 𝑚^∗𝜇_𝐸≥^∞_𝑗=1𝑚^∗𝜇_𝐸∧𝜇_𝐵𝑗+𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐴.(3.22) Using the semiadditivity, we have, 𝑚^∗𝜇_𝐸∧𝜇_𝐴=𝑚^∗∞_𝑗=1𝜇_𝐸∧𝜇_𝐵𝑗=𝑚^∗𝜇_𝐸∧^∞_𝑗=1𝜇_𝐵𝑗≤𝑚^∗𝜇_𝐸∧𝜇_𝐵𝑗.(3.23) From (3.22), we have 𝑚^∗𝜇_𝐸≥𝑚^∗𝜇_𝐸∧𝜇_𝐴+𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐴.(3.24) Hence, 𝜇_𝐴∈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.
Let 𝜇_𝐸=⋁^∞_𝑗=1𝜇_𝐴𝑗. From (3.22), we have 𝑚^∗∞_𝑗=1𝜇_𝐴𝑗≥^∞_𝑗=1𝑚^∗𝜇_𝐴𝑗.(3.25)
Step  6. We have 𝜎(𝐿)⊃𝜎(𝐿).
Let 𝜇_𝐴∈𝜎(𝐿) and 𝜇_𝐸≤𝜇_𝐴. Then, we must show the following inequality: 𝑚^∗𝜇_𝐸≥𝑚^∗𝜇_𝐸∧𝜇_𝐴+𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐴.(3.26)
If 𝜇_𝐸∈𝜎(𝐿), then 𝜇_𝐸∧𝜇_𝐴 and 𝜇_𝐸∧𝜇^𝑐_𝐴 are different and both of them belong to 𝜎(𝐿), (3.26) is obvious and since 𝑚^∗=𝑚, hence additive.
With  𝜇_𝐸≤𝜇_𝑋 and given 𝜀>0, 𝜎(𝐿), there is 𝜇_𝐸𝑗 which contains  𝜎(𝐿) such that we have 𝑚^∗𝜇_𝐸+𝜀>^∞_𝑗=1𝑚𝜇_𝐸𝑗.(3.27)
Now, from the equality 𝜇_𝐸𝑗=𝜇_𝐸𝑗∧𝜇_𝐴∨𝜇_𝐸𝑗∧𝜇^𝑐_𝐴(3.28) and from the Definition 2.1 and Theorem 2.3, we have the following equality: 𝑚𝜇_𝐸𝑗=𝑚𝜇_𝐸𝑗∧𝜇_𝐴+𝑚𝜇_𝐸𝑗∧𝜇^𝑐_𝐴.(3.29)
Therefore, we obtain the following: 𝜇_𝐸∧𝜇_𝐴≤^∞_𝑗=1𝜇_𝐸𝑗∧𝜇_𝐴,𝜇_𝐸∧𝜇^𝑐_𝐴≤^∞_𝑗=1𝜇_𝐸𝑗∧𝜇^𝑐_𝐴.(3.30) Using the monotonicity and semiadditivity, we obtain 𝑚^∗𝜇_𝐸∧𝜇_𝐴≤^∞_𝑗=1𝑚𝜇_𝐸𝑗∧𝜇_𝐴,𝑚^∗𝜇_𝐸∧𝜇^𝑐_𝐴≤^∞_𝑗=1𝑚𝜇_𝐸𝑗∧𝜇^𝑐_𝐴.(3.31) Using the sum of the inequalities (3.31), 𝑚^∗𝜇_𝐸∧𝜇_𝐴+𝑚^∗𝜇_𝐸𝑗∧𝜇^𝑐_𝐴≤^∞_𝑗=1𝑚^∗𝜇_𝐸𝑗<𝑚^∗𝜇_𝐸+𝜀(3.32) 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(𝜇_𝐸)=𝑚(𝜇_𝐸) for all 𝜇_𝐸∈𝜎(𝐿). We must show that 𝑚₁𝜇_𝐴=𝑚𝜇_𝐴.(3.33) Since 𝑚 is a finite 𝜎-lattice fuzzy measure, we can write 𝑋=^∞_𝑛=1𝜇_𝐸𝑛,𝜇_𝐸𝑛∈𝜎(𝐿),𝑛≠𝑘,𝜇_𝐸𝑛∧𝜇_𝐸𝑘=∅,𝑚𝜇_𝐸𝑛<∞;1≤𝑛<∞.(3.34) If 𝜇_𝐴∈𝜎(𝐿), then we have 𝑚𝜇_𝐵=^∞_𝑛=1𝑚𝜇_𝐴∧𝜇_𝐸𝑛,𝑚₁𝜇_𝐴=^∞_𝑛=1𝑚₁𝜇_𝐴∧𝜇_𝐸𝑛.(3.35) To prove the inequality (3.33), it suffices to show that 𝑚₁𝜇_𝐴=𝑚𝜇_𝐴,𝜇_𝐴∈𝜎𝐿,𝑚𝜇_𝐴<∞.(3.36)
Let  𝜇_𝐴∈𝜎(𝐿), 𝑚(𝜇_𝐴)<∞, and 𝜀>0 arbitrary. Then, we have 𝜇_𝐴≤^∞_𝑛=1𝜇_𝐸𝑛,for𝜇_𝐸𝑛∈𝜎(𝐿),1≤𝑛<∞,(3.37)𝑚^∞_𝑛=1𝜇_𝐸𝑛≤^∞_𝑛=1𝑚𝜇_𝐸𝑛<𝑚𝜇_𝐴+𝜀.(3.38) Since 𝑚₁(𝜇_𝐴)≤𝑚₁(⋁^∞_𝑛=1𝜇_𝐸𝑛)≤⋁^∞_𝑛=1𝑚₁(𝜇_𝐸𝑛)=⋁^∞_𝑛=1𝑚(𝜇_𝐸𝑛) and from (3.38), we get 𝑚₁𝜇_𝐴≤𝑚𝜇_𝐴.(3.39) Also, from (3.38), we can write 𝜇_𝐹=∨^∞_𝑛=1𝜇_𝐸𝑛∈𝜎(𝐿) for the sets 𝜇_𝐸𝑛. Therefore, 𝜇_𝐹 is 𝑚^∗ lattice fuzzy measurable. From the inequality 𝜇_𝐴≤𝜇_𝐹 and (3.38), 𝑚𝜇_𝐹=𝑚𝜇_𝐴+𝑚𝜇_𝐹−𝜇_𝐴,𝑚𝜇_𝐹−𝜇_𝐴=𝑚𝜇_𝐹−𝑚𝜇_𝐴<𝜀(3.40) are obtained.
From the equalities 𝑚(𝜇_𝐸)=𝑚(𝜇_𝐸) and 𝑚₁(𝜇_𝐹)=𝑚(𝜇_𝐹) for all 𝜇_𝐴∈𝜎(𝐿), we can write 𝑚𝜇_𝐴≤𝑚𝜇_𝐹=𝑚₁𝜇_𝐹=𝑚₁𝜇_𝐴+𝑚₁𝜇_𝐹−𝜇_𝐴≤𝑚₁𝜇_𝐴+𝑚𝜇_𝐹−𝜇_𝐴.(3.41) Therefore, from (3.41), 𝑚𝜇_𝐴≤𝑚₁𝜇_𝐴(3.42) 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 𝜎(𝐿)=^∞_𝑖=1𝐿_0𝑖,𝐿_1𝑖∶𝐿_0𝑖,𝐿_1𝑖⊂𝐿₀,𝐿₁.(3.43) For 𝜇_𝐴∈𝜎(𝐿),𝑚(𝜇_𝐴)=∞ if 𝜇_𝐴≠∅, and 𝑚(𝜇_𝐴)=𝐿₀ if 𝜇_𝐴=∅.
After all these, solution of application is clearly in the generalized Caratheodory extension theorem at property (v).