Abstract

The notion of ( šæ , š‘€ )-fuzzy šœŽ -algebras is introduced in the lattice value fuzzy set theory. It is a generalization of Klement's fuzzy šœŽ -algebras. In our definition of ( šæ , š‘€ )-fuzzy šœŽ -algebras, each šæ -fuzzy subset can be regarded as an šæ -measurable set to some degree.

1. Introduction and Preliminaries

In 1980, Klement established an axiomatic theory of fuzzy šœŽ -algebras in [1] in order to prepare a measure theory for fuzzy sets. In the definition of Klement's fuzzy šœŽ -algebra ( š‘‹ , šœŽ ) , šœŽ was defined as a crisp family of fuzzy subsets of a set š‘‹ satisfying certain set of axioms. In 1991, Biacino and Lettieri generalized Klement's fuzzy šœŽ -algebras to šæ -fuzzy setting [2].

In this paper, when both šæ and š‘€ are complete lattices, we define an ( šæ , š‘€ ) -fuzzy šœŽ -algebra on a nonempty set š‘‹ by means of a mapping šœŽ āˆ¶ šæ š‘‹ ā†’ š‘€ satisfying three axioms. Thus each šæ -fuzzy subset of š‘‹ can be regarded as an šæ -measurable set to some degree.

When šœŽ is an ( šæ , š‘€ ) -fuzzy šœŽ -algebra on š‘‹ , ( š‘‹ , šœŽ ) is called an ( šæ , š‘€ ) -fuzzy measurable space. An ( šæ , 2 ) -fuzzy šœŽ -algebra is also called an šæ - šœŽ -algebra. A Klement šœŽ -algebra can be viewed as a stratified [ 0 , 1 ] - šœŽ -algebra. A Biacino-Lettieri šæ - šœŽ -algebra can be viewed as a stratified šæ - šœŽ -algebra. A ( 2 , š‘€ ) -fuzzy šœŽ -algebra is also called an š‘€ -fuzzifying šœŽ -algebra. A crisp šœŽ -algebra can be regarded as a ( 2 , 2 ) -fuzzy šœŽ -algebra.

Throughout this paper, both šæ and š‘€ denote complete lattices, and šæ has an order-reversing involution ā€² . š‘‹ is a nonempty set. šæ š‘‹ is the set of all šæ -fuzzy sets (or šæ -sets for short) on š‘‹ . We often do not distinguish a crisp subset š“ of š‘‹ and its character function šœ’ š“ . The smallest element and the largest element in š‘€ are denoted by āŸ‚ š‘€ and āŠ¤ š‘€ , respectively.

The binary relation ā‰ŗ in š‘€ is defined as follows: for š‘Ž , š‘ āˆˆ š‘€ , š‘Ž ā‰ŗ š‘ if and only if for every subset š· āŠ† š‘€ , the relation š‘ ā©½ s u p š· always implies the existence of š‘‘ āˆˆ š· with š‘Ž ā©½ š‘‘ [3]. { š‘Ž āˆˆ š‘€ āˆ¶ š‘Ž ā‰ŗ š‘ } is called the greatest minimal family of š‘ in the sense of [4], denoted by š›½ ( š‘ ) . Moreover, for š‘ āˆˆ š‘€ , we define š›¼ ( š‘ ) = { š‘Ž āˆˆ š‘€ āˆ¶ š‘Ž ā‰ŗ š‘œ š‘ š‘ } . In a completely distributive lattice š‘€ , there exist š›¼ ( š‘ ) and š›½ ( š‘ ) for each š‘ āˆˆ š‘€ , and ā‹ ā‹€ š‘ = š›½ ( š‘ ) = š›¼ ( š‘ ) (see [4]).

In [4], Wang thought that š›½ ( 0 ) = { 0 } and š›¼ ( 1 ) = { 1 } . In fact, it should be that š›½ ( 0 ) = āˆ… and š›¼ ( 1 ) = āˆ… .

For a complete lattice šæ , š“ āˆˆ šæ š‘‹ and š‘Ž āˆˆ šæ , we use the following notation:

š“ [ š‘Ž ] = { š‘„ āˆˆ š‘‹ āˆ¶ š“ ( š‘„ ) ā©¾ š‘Ž } . ( 1 . 1 )

If šæ is completely distributive, then we can define

š“ [ š‘Ž ] = { š‘„ āˆˆ š‘‹ āˆ¶ š‘Ž ĀØ ĀØ āˆˆ š›¼ ( š“ ( š‘„ ) ) } . ( 1 . 2 )

Some properties of these cut sets can be found in [5ā€“10].

Theorem 1.1 (see [4]). Let š‘€ be a completely distributive lattice and { š‘Ž š‘– āˆ¶ š‘– āˆˆ Ī© } āŠ† š‘€ . Then (1) ā‹€ š›¼ ( š‘– āˆˆ Ī© š‘Ž š‘– ā‹ƒ ) = š‘– āˆˆ Ī© š›¼ ( š‘Ž š‘– ) , that is, š›¼ is an ā‹€ āˆ’ ā‹ƒ map;(2) ā‹ š›½ ( š‘– āˆˆ Ī© š‘Ž š‘– ā‹ƒ ) = š‘– āˆˆ Ī© š›½ ( š‘Ž š‘– ) , that is, š›½ is a union-preserving map. For š‘Ž āˆˆ šæ and š· āŠ† š‘‹ , we define two šæ -fuzzy sets š‘Ž ā‹€ š· and š‘Ž āˆØ š· as follows: īƒÆ ( š‘Ž āˆ§ š· ) ( š‘„ ) = š‘Ž , š‘„ āˆˆ š· ; 0 , š‘„ ĀØ ĀØ īƒÆ āˆˆ š· . ( š‘Ž āˆØ š· ) ( š‘„ ) = 1 , š‘„ āˆˆ š· ; š‘Ž , š‘„ ĀØ ĀØ āˆˆ š· . ( 1 . 3 ) Then for each šæ -fuzzy set š“ in šæ š‘‹ , it follows that ī˜ š“ = š‘Ž āˆˆ šæ ī€· š‘Ž āˆ§ š“ [ š‘Ž ] ī€ø . ( 1 . 4 )

Theorem 1.2 (see [5, 7, 10]). If šæ is completely distributive, then for each šæ -fuzzy set š“ in šæ š‘‹ , we have (1) ā‹ š“ = š‘Ž āˆˆ šæ ā‹€ š“ ( š‘Ž [ š‘Ž ] ā‹€ ) = š‘Ž āˆˆ šæ ( š‘Ž āˆØ š“ [ š‘Ž ] ) ; (2) f o r a l l š‘Ž āˆˆ šæ , š“ [ š‘Ž ] = ā‹‚ š‘ āˆˆ š›½ ( š‘Ž ) š“ [ š‘ ] ; (3) f o r a l l š‘Ž āˆˆ šæ , š“ [ š‘Ž ] = ā‹‚ š‘Ž āˆˆ š›¼ ( š‘ ) š“ [ š‘ ] . For a family of šæ -fuzzy sets { š“ š‘– āˆ¶ š‘– āˆˆ Ī© } in šæ š‘‹ , it is easy to see that īƒ© ī— š‘– āˆˆ Ī© š“ š‘– īƒŖ [ š‘Ž ] = ī™ š‘– āˆˆ Ī© ī€· š“ š‘– ī€ø [ š‘Ž ] . ( 1 . 5 ) If šæ is completely distributive, then it follows [7] that īƒ© ī— š‘– āˆˆ Ī© š“ š‘– īƒŖ [ š‘Ž ] = ī™ š‘– āˆˆ Ī© ī€· š“ š‘– ī€ø [ š‘Ž ] . ( 1 . 6 )

Definition 1.3. Let š‘‹ be a nonempty set. A subset šœŽ of [ 0 , 1 ] š‘‹ is called a Klement fuzzy šœŽ -algebra if it satisfies the following three conditions:(1)for any constant fuzzy set š›¼ , š›¼ āˆˆ šœŽ ;(2)for any š“ āˆˆ [ 0 , 1 ] š‘‹ , 1 āˆ’ š“ āˆˆ šœŽ ;(3)for any { š“ š‘› āˆ¶ š‘› āˆˆ ā„• } āŠ† šœŽ , ā‹ š‘› āˆˆ ā„• š“ š‘› āˆˆ šœŽ .The fuzzy sets in šœŽ are called fuzzy measurable sets, and the pair ( š‘‹ , šœŽ ) a fuzzy measurable space.

Definition 1.4. Let šæ be a complete lattice with an order-reversing involution ī…ž and š‘‹ a nonempty set. A subset šœŽ of šæ š‘‹ is called an šæ - šœŽ -algebra if it satisfies the following three conditions:(1)for any š‘Ž āˆˆ šæ , constant šæ -fuzzy set š‘Ž ā‹€ šœ’ š‘‹ āˆˆ šœŽ ;(2)for any š“ āˆˆ šæ š‘‹ , š“ ī…ž āˆˆ šœŽ ;(3)for any { š“ š‘› āˆ¶ š‘› āˆˆ ā„• } āŠ† šœŽ , ā‹ š‘› āˆˆ ā„• š“ š‘› āˆˆ šœŽ .The šæ -fuzzy sets in šœŽ are called šæ -measurable sets, and the pair ( š‘‹ , šœŽ ) an šæ -measurable space.

2. ( šæ , š‘€ ) -Fuzzy šœŽ -Algebras

L. Biacino and A. Lettieri defined that an šæ - šœŽ -algebra šœŽ is a crisp subset of šæ š‘‹ . Now we consider an š‘€ -fuzzy subset šœŽ of šæ š‘‹ .

Definition. Let š‘‹ be a nonempty set. A mapping šœŽ āˆ¶ šæ š‘‹ ā†’ š‘€ is called an ( šæ , š‘€ ) -fuzzy šœŽ -algebra if it satisfies the following three conditions: (LMS1) šœŽ ( šœ’ āˆ… ) = āŠ¤ š‘€ ;(LMS2) for any š“ āˆˆ šæ š‘‹ , šœŽ ( š“ ) = šœŽ ( š“ ī…ž ) ;(LMS3) for any { š“ š‘› āˆ¶ š‘› āˆˆ ā„• } āŠ† šæ š‘‹ , ā‹ šœŽ ( š‘› āˆˆ ā„• š“ š‘› ā‹€ ) ā‰„ š‘› āˆˆ ā„• šœŽ ( š“ š‘› ) .An ( šæ , š‘€ ) -fuzzy šœŽ -algebra šœŽ is said to be stratified if and only if it satisfies the following condition:
(LMS1)* ā€‰ā€‰ āˆ€ š‘Ž āˆˆ šæ , ā‹€ šœ’ šœŽ ( š‘Ž š‘‹ ) = āŠ¤ š‘€ .
If šœŽ is an ( šæ , š‘€ ) -fuzzy šœŽ -algebra, then ( š‘‹ , šœŽ ) is called an ( šæ , š‘€ ) -fuzzy measurable space.
An ( šæ , šŸ ) -fuzzy šœŽ -algebra is also called an šæ - šœŽ -algebra, and an ( šæ , šŸ ) -fuzzy measurable space is also called an šæ -measurable space.
A ( šŸ , š‘€ ) -fuzzy šœŽ -algebra is also called an š‘€ -fuzzifying šœŽ -algebra, and a ( šŸ , š‘€ ) -fuzzy measurable space is also called an š‘€ -fuzzifying measurable space.
Obviously a crisp measurable space can be regarded as a ( šŸ , šŸ ) -fuzzy measurable space.
If šœŽ is an ( šæ , š‘€ ) -fuzzy šœŽ -algebra, then šœŽ ( š“ ) can be regarded as the degree to which š“ is an šæ -measurable set.

Remark 2.2. If a subset šœŽ of šæ š‘‹ is regarded as a mapping šœŽ āˆ¶ šæ š‘‹ ā†’ šŸ , then šœŽ is an šæ - šœŽ -algebra if and only if it satisfies the following conditions:(LS1) šœ’ āˆ… āˆˆ šœŽ ;(LS2) š“ āˆˆ šœŽ ā‡’ š“ ī…ž āˆˆ šœŽ ;(LS3) for any { š“ š‘› āˆ¶ š‘› āˆˆ ā„• } āŠ† šœŽ , ā‹ š‘› āˆˆ ā„• š“ š‘› āˆˆ šœŽ . Thus we easily see that a Klement šœŽ -algebra is exactly a stratified [ 0 , 1 ] - šœŽ -algebra, and a Biacino-Lettieri šæ - šœŽ -algebra is exactly a stratified šæ - šœŽ -algebra.
Moreover, when šæ = šŸ , a mapping šœŽ āˆ¶ šŸ š‘‹ ā†’ š‘€ is an š‘€ -fuzzifying šœŽ -algebra if and only if it satisfies the following conditions:
(MS1) šœŽ ( āˆ… ) = āŠ¤ š‘€ ;(MS2) for any š“ āˆˆ šŸ š‘‹ , šœŽ ( š“ ) = šœŽ ( š“ ī…ž ) ;(MS3) for any { š“ š‘› āˆ¶ š‘› āˆˆ ā„• } āŠ† 2 š‘‹ , ā‹ šœŽ ( š‘› āˆˆ ā„• š“ š‘› ā‹€ ) ā‰„ š‘› āˆˆ ā„• šœŽ ( š“ š‘› ) .

Example 2.3. Letā€‰ā€‰ ( š‘‹ , šœŽ ) be a crisp measurable space. Define šœ’ šœŽ āˆ¶ šŸ š‘‹ ā†’ [ 0 , 1 ] by šœ’ šœŽ īƒÆ ( š“ ) = 1 , š“ āˆˆ šœŽ ; 0 , š“ ĀØ ĀØ āˆˆ šœŽ . ( 2 . 1 ) Then it is easy to prove that ( š‘‹ , šœ’ šœŽ ) is a [ 0 , 1 ] -fuzzifying measurable space.

Example 2.4. Let š‘‹ be a nonempty set and šœŽ āˆ¶ šŸ š‘‹ ā†’ [ 0 , 1 ] a mapping defined by īƒÆ šœŽ ( š“ ) = 1 , š“ āˆˆ { āˆ… , š‘‹ } ; 0 . 5 , š“ ĀØ ĀØ āˆˆ { āˆ… , š‘‹ } . ( 2 . 2 ) Then it is easy to prove that ( š‘‹ , šœŽ ) is a [ 0 , 1 ] -fuzzifying measurable space. If š“ āˆˆ šŸ š‘‹ with š“ ĀØ ĀØ āˆˆ { āˆ… , š‘‹ } , then 0.5 is the degree to which š“ is measurable.

Example 2.5. Let š‘‹ be a nonempty set and šœŽ āˆ¶ [ 0 , 1 ] š‘‹ ā†’ [ 0 , 1 ] a mapping defined by īƒÆ ī€½ šœ’ šœŽ ( š“ ) = 1 , š“ āˆˆ āˆ… , šœ’ š‘‹ ī€¾ ; 0 . 5 , š“ ĀØ ĀØ āˆˆ ī€½ šœ’ āˆ… , šœ’ š‘‹ ī€¾ . ( 2 . 3 ) Then it is easy to prove that ( š‘‹ , šœŽ ) is a ( [ 0 , 1 ] , [ 0 , 1 ] ) -fuzzy measurable space. If š“ āˆˆ [ 0 , 1 ] š‘‹ with š“ ĀØ ĀØ āˆˆ { šœ’ āˆ… , šœ’ š‘‹ } , then 0.5 is the degree to which š“ is [ 0 , 1 ] -measurable.

Proposition 2.6. Let ( š‘‹ , šœŽ ) be an ( šæ , š‘€ ) -fuzzy measurable spaces. Then for any { š“ š‘› āˆ¶ š‘› āˆˆ ā„• } āŠ† šæ š‘‹ , ā‹€ šœŽ ( š‘› āˆˆ ā„• š“ š‘› ā‹€ ) ā‰„ š‘› āˆˆ ā„• šœŽ ( š“ š‘› ) .

Proof. This can be proved from the following fact: šœŽ īƒ© ī— š‘› āˆˆ ā„• š“ š‘› īƒŖ īƒ© ī˜ = šœŽ š‘› āˆˆ ā„• ī€· š“ š‘› ī€ø ī…ž īƒŖ ā‰„ ī— š‘› āˆˆ ā„• šœŽ ī‚€ ī€· š“ š‘› ī€ø ī…ž ī‚ = ī— š‘› āˆˆ ā„• šœŽ ī€· š“ š‘› ī€ø . ( 2 . 4 ) The next two theorems give characterizations of an ( šæ , š‘€ ) -fuzzy šœŽ -algebra.

Theorem 2.7. A mapping šœŽ āˆ¶ šæ š‘‹ ā†’ š‘€ is an ( šæ , š‘€ ) -fuzzy šœŽ -algebra if and only if for each š‘Ž āˆˆ š‘€ ā§µ { āŸ‚ š‘€ } , šœŽ [ š‘Ž ] is an šæ - šœŽ -algebra.

Proof. The proof is obvious and is omitted.

Corollary 2.8. A mapping šœŽ āˆ¶ šŸ š‘‹ ā†’ š‘€ is an š‘€ -fuzzifying šœŽ -algebra if and only if for each š‘Ž āˆˆ š‘€ ā§µ { āŸ‚ š‘€ } , šœŽ [ š‘Ž ] is a šœŽ -algebra.

Theorem 2.9. If š‘€ is completely distributive, then a mapping šœŽ āˆ¶ šæ š‘‹ ā†’ š‘€ is an ( šæ , š‘€ ) -fuzzy šœŽ -algebra if and only if for each š‘Ž āˆˆ š›¼ ( āŸ‚ š‘€ ) , šœŽ [ š‘Ž ] is an šæ - šœŽ -algebra.

Proof. Necessity.. Suppose that šœŽ āˆ¶ šæ š‘‹ ā†’ š‘€ is an ( šæ , š‘€ ) -fuzzy šœŽ -algebra and š‘Ž āˆˆ š›¼ ( āŸ‚ š‘€ ) . Now we prove that šœŽ [ š‘Ž ] is an šæ - šœŽ -algebra.(LS1) By šœŽ ( šœ’ āˆ… ) = āŠ¤ š‘€ and š›¼ ( āŠ¤ š‘€ ) = āˆ… , we know that š‘Ž ĀØ ĀØ āˆˆ š›¼ ( šœŽ ( šœ’ āˆ… ) ) ; this implies that šœ’ āˆ… āˆˆ šœŽ [ š‘Ž ] .(LS2) If š“ āˆˆ šœŽ [ š‘Ž ] , then š‘Ž ĀØ ĀØ āˆˆ š›¼ ( šœŽ ( š“ ) ) = š›¼ ( šœŽ ( š“ ī…ž ) ) ; this shows that š“ ī…ž āˆˆ šœŽ [ š‘Ž ] .(LS3) If { š“ š‘– āˆ¶ š‘– āˆˆ Ī© } āŠ† šœŽ [ š‘Ž ] , then f o r a l l š‘– āˆˆ Ī© , š‘Ž ĀØ ĀØ āˆˆ š›¼ ( šœŽ ( š“ š‘– ) ) . Hence š‘Ž ĀØ ĀØ āˆˆ ā‹ƒ š‘– āˆˆ Ī© š›¼ ( šœŽ ( š“ š‘– ) ) . By ā‹ šœŽ ( š‘– āˆˆ Ī© š“ š‘– ā‹€ ) ā‰„ š‘– āˆˆ Ī© šœŽ ( š“ š‘– ) , we know that š›¼ īƒ© šœŽ īƒ© ī˜ š‘– āˆˆ Ī© š“ š‘– īƒ© ī— īƒŖ īƒŖ āŠ† š›¼ š‘– āˆˆ Ī© šœŽ ī€· š“ š‘– ī€ø īƒŖ = īš š‘– āˆˆ Ī© š›¼ ī€· šœŽ ī€· š“ š‘– . ī€ø ī€ø ( 2 . 5 ) This shows that š‘Ž ĀØ ĀØ ā‹ āˆˆ š›¼ ( šœŽ ( š‘– āˆˆ Ī© š“ š‘– ) ) . Therefore, ā‹ š‘– āˆˆ Ī© š“ š‘– āˆˆ šœŽ [ š‘Ž ] . The proof is completed.

Corollary 2.10. If š‘€ is completely distributive, then a mapping šœŽ āˆ¶ šŸ š‘‹ ā†’ š‘€ is an š‘€ -fuzzifying šœŽ -algebra if and only if for each š‘Ž āˆˆ š›¼ ( āŸ‚ š‘€ ) , šœŽ [ š‘Ž ] is a šœŽ -algebra.

Now we consider the conditions that a family of šæ - šœŽ -algebras forms an ( šæ , š‘€ ) -fuzzy šœŽ -algebra. By Theorem 1.2, we can obtain the following result.

Corollary 2.11. If š‘€ is completely distributive, and šœŽ is an ( šæ , š‘€ ) -fuzzy šœŽ -algebra, then (1) šœŽ [ š‘ ] āŠ† šœŽ [ š‘Ž ] for any š‘Ž , š‘ āˆˆ š‘€ ā§µ { āŸ‚ š‘€ } with š‘Ž āˆˆ š›½ ( š‘ ) ;(2) šœŽ [ š‘ ] āŠ† šœŽ [ š‘Ž ] for any š‘Ž , š‘ āˆˆ š›¼ ( āŸ‚ š‘€ ) with š‘ āˆˆ š›¼ ( š‘Ž ) .

Theorem 2.12. Let š‘€ be completely distributive, and let { šœŽ š‘Ž āˆ¶ š‘Ž āˆˆ š›¼ ( āŸ‚ š‘€ ) } be a family of šæ - šœŽ -algebras. If šœŽ š‘Ž = ā‹‚ { šœŽ š‘ āˆ¶ š‘Ž āˆˆ š›¼ ( š‘ ) } for all š‘Ž āˆˆ š›¼ ( āŸ‚ š‘€ ) , then there exists an ( šæ , š‘€ ) -fuzzy šœŽ -algebra šœŽ such that šœŽ [ š‘Ž ] = šœŽ š‘Ž .

Proof. Suppose that šœŽ š‘Ž = ā‹‚ { šœŽ š‘ āˆ¶ š‘Ž āˆˆ š›¼ ( š‘ ) } for all š‘Ž āˆˆ š›¼ ( āŸ‚ š‘€ ) . Define šœŽ āˆ¶ šæ š‘‹ ā†’ š‘€ by ī— šœŽ ( š“ ) = š‘Ž āˆˆ š‘€ ( š‘Ž āˆØ šœŽ š‘Ž ī— ( š“ ) ) = { š‘Ž āˆˆ š‘€ āˆ¶ š“ ĀØ ĀØ āˆˆ šœŽ š‘Ž } . ( 2 . 6 ) By Theorem 1.2, we can obtain that šœŽ [ š‘Ž ] = šœŽ š‘Ž .

Corollary 2.13. Let š‘€ be completely distributive, and let { šœŽ š‘Ž āˆ¶ š‘Ž āˆˆ š›¼ ( āŸ‚ š‘€ ) } be a family of šœŽ -algebras. If šœŽ š‘Ž = ā‹‚ { šœŽ š‘ āˆ¶ š‘Ž āˆˆ š›¼ ( š‘ ) } for all š‘Ž āˆˆ š›¼ ( āŸ‚ š‘€ ) , then there exists an š‘€ -fuzzifying šœŽ -algebra šœŽ such that šœŽ [ š‘Ž ] = šœŽ š‘Ž .

Theorem 2.14. Let š‘€ be completely distributive, and let { šœŽ š‘Ž āˆ¶ š‘Ž āˆˆ š‘€ ā§µ { āŸ‚ š‘€ } } be a family of šæ - šœŽ -algebra. If šœŽ š‘Ž = ā‹‚ { šœŽ š‘ āˆ¶ š‘ āˆˆ š›½ ( š‘Ž ) } for all š‘Ž āˆˆ š‘€ ā§µ { āŸ‚ š‘€ } , then there exists an ( šæ , š‘€ ) -fuzzy šœŽ -algebra šœŽ such that šœŽ [ š‘Ž ] = šœŽ š‘Ž .

Proof. Suppose that šœŽ š‘Ž = ā‹‚ { šœŽ š‘ āˆ¶ š‘ āˆˆ š›½ ( š‘Ž ) } for all š‘Ž āˆˆ š‘€ ā§µ { āŸ‚ š‘€ } . Define šœŽ āˆ¶ šæ š‘‹ ā†’ š‘€ by ī˜ šœŽ ( š“ ) = š‘Ž āˆˆ š‘€ ī€· š‘Ž āˆ§ šœŽ š‘Ž ī€ø = ī˜ ī€½ ( š“ ) š‘Ž āˆˆ š‘€ āˆ¶ š“ āˆˆ šœŽ š‘Ž ī€¾ . ( 2 . 7 ) By Theorem 1.2, we can obtain šœŽ [ š‘Ž ] = šœŽ š‘Ž .

Corollary 2.15. Let š‘€ be completely distributive, and let { šœŽ š‘Ž āˆ¶ š‘Ž āˆˆ š‘€ ā§µ { āŸ‚ š‘€ } } be a family of šœŽ -algebra. If šœŽ š‘Ž = ā‹‚ { šœŽ š‘ āˆ¶ š‘ āˆˆ š›½ ( š‘Ž ) } for all š‘Ž āˆˆ š‘€ ā§µ { āŸ‚ š‘€ } , then there exists an š‘€ -fuzzifying šœŽ -algebra šœŽ such that šœŽ [ š‘Ž ] = šœŽ š‘Ž .

Theorem 2.16. Let { šœŽ š‘– āˆ¶ š‘– āˆˆ Ī© } be a family of ( šæ , š‘€ ) -fuzzy šœŽ -algebra on š‘‹ . Then ā‹€ š‘– āˆˆ Ī© šœŽ š‘– is an ( šæ , š‘€ ) -fuzzy šœŽ -algebra on š‘‹ , where ā‹€ š‘– āˆˆ Ī© šœŽ š‘– āˆ¶ šæ š‘‹ ā†’ š‘€ is defined by ( ā‹€ š‘– āˆˆ Ī© šœŽ š‘– ā‹€ ) ( š“ ) = š‘– āˆˆ Ī© šœŽ š‘– ( š“ ) .

Proof. This is straightforward.

3. ( šæ , š‘€ ) -Fuzzy Measurable Functions

In this section, we will generalize the notion of measurable functions to fuzzy setting.

Theorem 3.1. Let ( š‘Œ , šœ ) be an ( šæ , š‘€ ) -fuzzy measurable space and š‘“ āˆ¶ š‘‹ ā†’ š‘Œ a mapping. Define a mapping š‘“ ā† šæ ( šœ ) āˆ¶ šæ š‘‹ ā†’ š‘€ by f o r a l l š“ āˆˆ šæ š‘‹ , š‘“ ā† šæ ī˜ ī€½ ( šœ ) ( š“ ) = šœ ( šµ ) āˆ¶ š‘“ ā† šæ ī€¾ , ( šµ ) = š“ w h e r e āˆ€ š‘„ āˆˆ š‘‹ , š‘“ ā† šæ ( šµ ) ( š‘„ ) = šµ ( š‘“ ( š‘„ ) ) . ( 3 . 1 ) Then ( š‘‹ , š‘“ ā† šæ ( šœ ) ) is an ( šæ , š‘€ ) -fuzzy measurable space.

Proof. (LMS1) holds from the following equality: š‘“ ā† šæ ī€· šœ’ ( šœ ) āˆ… ī€ø = ī˜ ī€½ šœ ( šµ ) āˆ¶ š‘“ ā† šæ ( šµ ) = šœ’ āˆ… ī€¾ ī€· šœ’ = šœ āˆ… ī€ø = āŠ¤ š‘€ . ( 3 . 2 )
(LMS2) can be shown from the following fact: f o r a l l š“ āˆˆ šæ š‘‹ , š‘“ ā† šæ ī˜ ī€½ ( šœ ) ( š“ ) = šœ ( šµ ) āˆ¶ š‘“ ā† šæ ī€¾ = ī˜ ī€½ šœ ī€· šµ ( šµ ) = š“ ī…ž ī€ø āˆ¶ š‘“ ā† šæ ī€· šµ ī…ž ī€ø = š‘“ ā† šæ ( šµ ) ī…ž = š“ ī…ž ī€¾ = š‘“ ā† šæ ī€· š“ ( šœ ) ī…ž ī€ø . ( 3 . 3 )
(LMS3) for any { š“ š‘› āˆ¶ š‘› āˆˆ ā„• } āŠ† šæ š‘‹ , by š‘“ ā† šæ īƒ© ī˜ ( šœ ) š‘› āˆˆ ā„• š“ š‘› īƒŖ = ī˜ īƒÆ šœ ( šµ ) āˆ¶ š‘“ ā† šæ ī˜ ( šµ ) = š‘› āˆˆ ā„• š“ š‘› īƒ° ā‰„ ī˜ īƒÆ šœ īƒ© ī˜ š‘› āˆˆ ā„• šµ š‘› īƒŖ āˆ¶ š‘“ ā† šæ ī€· šµ š‘› ī€ø = š“ š‘› īƒ° ā‰„ ī— š‘› āˆˆ ā„• š‘“ ā† šæ ( ī€· š“ šœ ) š‘› ī€ø ( 3 . 4 ) w e c a n p r o v e ( L M S 3 ) .

Definition 3.2. Let ( š‘‹ , šœŽ ) and ( š‘Œ , šœ ) be ( šæ , š‘€ ) -fuzzy measurable spaces. A mapping š‘“ āˆ¶ š‘‹ ā†’ š‘Œ is called ( šæ , š‘€ ) -fuzzy measurable if šœŽ ( š‘“ ā† šæ ( šµ ) ) ā‰„ šœ ( šµ ) for all šµ āˆˆ šæ š‘Œ .
An ( šæ , šŸ ) -fuzzy measurable mapping is called an šæ -measurable mapping, and a ( šŸ , š‘€ ) -fuzzy measurable mapping is called an š‘€ -fuzzifying measurable mapping.

Obviously a Klement fuzzy measurable mapping can be viewed as an [ 0 , 1 ] -measurable mapping.

The following theorem gives a characterization of ( šæ , š‘€ ) -fuzzy measurable mappings.

Theorem 3.3. Let ( š‘‹ , šœŽ ) and ( š‘Œ , šœ ) be two ( šæ , š‘€ ) -fuzzy measurable spaces. A mapping š‘“ āˆ¶ š‘‹ ā†’ š‘Œ is ( šæ , š‘€ ) -fuzzy measurable if and only if š‘“ ā† šæ ( šœ ) ( š“ ) ā‰¤ šœŽ ( š“ ) for all š“ āˆˆ šæ š‘‹ .

Proof. Necessity.. If š‘“ āˆ¶ š‘‹ ā†’ š‘Œ is ( šæ , š‘€ ) -fuzzy measurable, then šœŽ ( š‘“ ā† šæ ( šµ ) ) ā‰„ šœ ( šµ ) for all šµ āˆˆ šæ š‘Œ . Hence for all šµ āˆˆ šæ š‘Œ , we have š‘“ ā† šæ ī˜ ī€½ ( šœ ) ( š“ ) = šœ ( šµ ) āˆ¶ š‘“ ā† šæ ī€¾ ā‰¤ ī˜ ī€½ šœŽ ī€· š‘“ ( šµ ) = š“ ā† šæ ī€ø ( šµ ) āˆ¶ š‘“ ā† šæ ī€¾ ( šµ ) = š“ = šœŽ ( š“ ) . ( 3 . 5 ) Sufficiency.. If š‘“ ā† šæ ( šœ ) ( š“ ) ā‰¤ šœŽ ( š“ ) for all š“ āˆˆ šæ š‘‹ , then šœ ( šµ ) ā‰¤ š‘“ ā† šæ ( šœ ) ( š‘“ ā† šæ ( šµ ) ) ā‰¤ šœŽ ( š‘“ ā† šæ ( šµ ) ) for all šµ āˆˆ šæ š‘Œ ; this shows that š‘“ āˆ¶ š‘‹ ā†’ š‘Œ is ( šæ , š‘€ ) -fuzzy measurable.

The next three theorems are trivial.

Theorem 3.4. If š‘“ āˆ¶ ( š‘‹ , šœŽ ) ā†’ ( š‘Œ , šœ ) and š‘“ āˆ¶ ( š‘Œ , šœ ) ā†’ ( š‘ , šœŒ ) are ( šæ , š‘€ ) -fuzzy measurable, then š‘” āˆ˜ š‘“ āˆ¶ ( š‘‹ , šœŽ ) ā†’ ( š‘ , šœŒ ) is ( šæ , š‘€ ) -fuzzy measurable.

Theorem 3.5. Let ( š‘‹ , šœŽ ) and ( š‘Œ , šœ ) be ( šæ , š‘€ ) -fuzzy measurable spaces. Then a mapping š‘“ āˆ¶ ( š‘‹ , šœŽ ) ā†’ ( š‘Œ , šœ ) is ( šæ , š‘€ ) -fuzzy measurable if and only if š‘“ āˆ¶ ( š‘‹ , šœŽ [ š‘Ž ] ) ā†’ ( š‘Œ , šœ [ š‘Ž ] ) is šæ -measurable for any š‘Ž āˆˆ š‘€ ā§µ { āŸ‚ š‘€ } .

Theorem 3.6. Let š‘€ be completely distributive, and let ( š‘‹ , šœŽ ) and ( š‘Œ , šœ ) be ( šæ , š‘€ ) -fuzzy measurable spaces. Then a mapping š‘“ āˆ¶ ( š‘‹ , šœŽ ) ā†’ ( š‘Œ , šœ ) is ( šæ , š‘€ ) -fuzzy measurable if and only if š‘“ āˆ¶ ( š‘‹ , šœŽ [ š‘Ž ] ) ā†’ ( š‘Œ , šœ [ š‘Ž ] ) is šæ -measurable for any š‘Ž āˆˆ š›¼ ( āŸ‚ š‘€ ) .

Corollary 3.7. Let ( š‘‹ , šœŽ ) and ( š‘Œ , šœ ) be š‘€ -fuzzifying measurable spaces. Then a mapping š‘“ āˆ¶ ( š‘‹ , šœŽ ) ā†’ ( š‘Œ , šœ ) is š‘€ -fuzzifying measurable if and only if š‘“ āˆ¶ ( š‘‹ , šœŽ [ š‘Ž ] ) ā†’ ( š‘Œ , šœ [ š‘Ž ] ) is measurable for any š‘Ž āˆˆ š‘€ ā§µ { āŸ‚ š‘€ } .

Corollary 3.8. Let š‘€ be completely distributive, and let ( š‘‹ , šœŽ ) and ( š‘Œ , šœ ) be š‘€ -fuzzifying measurable spaces. Then a mapping š‘“ āˆ¶ ( š‘‹ , šœŽ ) ā†’ ( š‘Œ , šœ ) is š‘€ -fuzzifying measurable if and only if š‘“ āˆ¶ ( š‘‹ , šœŽ [ š‘Ž ] ) ā†’ ( š‘Œ , šœ [ š‘Ž ] ) is measurable for any š‘Ž āˆˆ š›¼ ( āŸ‚ š‘€ ) .

4. ( š¼ , š¼ ) -Fuzzy šœŽ -Algebras Generated by š¼ -Fuzzifying šœŽ -Algebras

In this section, ā„¬ will be used to denote the šœŽ -algebra of Borel subsets of š¼ = [ 0 , 1 ] .

Theorem 4.1. Let ( š‘‹ , šœŽ ) be an š¼ -fuzzifying measurable space. Define a mapping šœ ( šœŽ ) āˆ¶ š¼ š‘‹ ā†’ š¼ by ī— šœ ( šœŽ ) ( š“ ) = šµ āˆˆ ā„¬ šœŽ ī€· š“ āˆ’ 1 ī€ø . ( šµ ) ( 4 . 1 ) Then šœ ( šœŽ ) is a stratified ( š¼ , š¼ ) -fuzzy šœŽ -algebra, which is said to be the ( š¼ , š¼ ) -fuzzy šœŽ -algebra generated by šœŽ .

Proof. (LMS1) For any šµ āˆˆ ā„¬ and for any š‘Ž āˆˆ š¼ , if š‘Ž āˆˆ šµ , then ā‹€ šœ’ ( š‘Ž š‘‹ ) āˆ’ 1 ( šµ ) = š‘‹ ; if š‘Ž ĀØ ĀØ āˆˆ šµ , then ā‹€ šœ’ ( š‘Ž š‘‹ ) āˆ’ 1 ( šµ ) = āˆ… . However, we have that ā‹€ šœ’ šœŽ ( ( š‘Ž š‘‹ ) āˆ’ 1 ( šµ ) ) = 1 . This shows that ā‹€ šœ’ šœ ( šœŽ ) ( š‘Ž š‘‹ ) = 1 .
(LMS2) f o r a l l š“ āˆˆ š¼ š‘‹ and f o r a l l šµ āˆˆ ā„¬ , we have ī€· š“ šœ ( šœŽ ) ī…ž ī€ø = ī— šµ āˆˆ ā„¬ šœŽ ī€· ( 1 āˆ’ š“ ) āˆ’ 1 ī€ø = ī— ( šµ ) šµ āˆˆ ā„¬ šœŽ = ī— ( { š‘„ āˆˆ š‘‹ āˆ¶ 1 āˆ’ š“ ( š‘„ ) āˆˆ šµ } ) šµ āˆˆ ā„¬ šœŽ ( { š‘„ āˆˆ š‘‹ āˆ¶ āˆƒ š‘ āˆˆ šµ , s . t = ī— . š“ ( š‘„ ) = 1 āˆ’ š‘ } ) šµ āˆˆ ā„¬ šœŽ ī€· š“ āˆ’ 1 ī€ø ( šµ ) = šœ ( šœŽ ) ( š“ ) . ( 4 . 2 )
(LMS3) for any { š“ š‘› āˆ¶ š‘› āˆˆ ā„• } āŠ† šæ š‘‹ and f o r a l l šµ āˆˆ ā„¬ , by šœ īƒ© ī˜ ( šœŽ ) š‘› āˆˆ ā„• š“ š‘› īƒŖ = ī— šµ āˆˆ ā„¬ šœŽ āŽ› āŽœ āŽœ āŽ īƒ© ī˜ š‘› āˆˆ ā„• š“ š‘› īƒŖ āˆ’ 1 āŽž āŽŸ āŽŸ āŽ  = ī— ( šµ ) šµ āˆˆ ā„¬ šœŽ īƒ© īš š‘› āˆˆ ā„• š“ š‘› āˆ’ 1 īƒŖ ā‰„ ī— ( šµ ) šµ āˆˆ ā„¬ ī— š‘› āˆˆ ā„• šœŽ ī€· š“ š‘› āˆ’ 1 ī€ø = ī— ( šµ ) š‘› āˆˆ ā„• ī— šµ āˆˆ ā„¬ šœŽ ī€· š“ š‘› āˆ’ 1 ī€ø = ī— ( šµ ) š‘› āˆˆ ā„• ī€· š“ šœ ( šœŽ ) š‘› ī€ø , ( 4 . 3 )
we obtain ā‹ šœ ( šœŽ ) ( š‘› āˆˆ ā„• š“ š‘› ā‹€ ) ā‰„ š‘› āˆˆ ā„• šœ ( šœŽ ) ( š“ š‘› ) .

Corollary 4.2. Let ( š‘‹ , šœŽ ) be a measurable space. Define a subset šœ ( šœŽ ) āŠ† š¼ š‘‹ ( can be viewed as a mapping šœ ( šœŽ ) āˆ¶ š¼ š‘‹ ā†’ šŸ ) by ī€½ šœ ( šœŽ ) = š“ āˆˆ š¼ š‘‹ āˆ¶ āˆ€ šµ āˆˆ ā„¬ , š“ āˆ’ 1 ī€¾ . ( šµ ) āˆˆ šœŽ ( 4 . 4 ) Then šœ ( šœŽ ) is a stratified š¼ - šœŽ -algebra.

From Corollary 4.2, we see that the functor šœ in Theorem 4.1 is a generalization of Klement functor šœ .

Theorem 4.3. Let ( š‘‹ , šœŽ ) and ( š‘Œ , šœ ) be two š¼ -fuzzifying measurable spaces, and š‘“ āˆ¶ š‘‹ ā†’ š‘Œ is a map. Then š‘“ āˆ¶ ( š‘‹ , šœŽ ) ā†’ ( š‘Œ , šœ ) is š¼ -fuzzifying measurable if and only if š‘“ āˆ¶ ( š‘‹ , šœ ( šœŽ ) ) ā†’ ( š‘Œ , šœ ( šœ ) ) is ( š¼ , š¼ ) -fuzzy measurable.

Proof. Necessity. Suppose that š‘“ āˆ¶ ( š‘‹ , šœŽ ) ā†’ ( š‘Œ , šœ ) is š¼ -fuzzifying measurable. Then šœŽ ( š‘“ āˆ’ 1 ( š“ ) ) ā‰„ šœ ( š“ ) for any š“ āˆˆ šŸ š‘‹ . In order to prove that š‘“ āˆ¶ ( š‘‹ , šœ ( šœŽ ) ) ā†’ ( š‘Œ , šœ ( šœ ) ) is ( š¼ , š¼ ) -fuzzy measurable, we need to prove that šœ ( šœŽ ) ( š‘“ ā† šæ ( š“ ) ) ā‰„ šœ ( šœ ) ( š“ ) for any š“ āˆˆ š¼ š‘‹ .
In fact, for any š“ āˆˆ š¼ š‘‹ , by ī€· š‘“ šœ ( šœŽ ) ā† šæ ī€ø = ī— ( š“ ) šµ āˆˆ ā„¬ šœŽ ī‚€ ī€· š‘“ ā† šæ ī€ø ( š“ ) āˆ’ 1 ī‚ = ī— ( šµ ) šµ āˆˆ ā„¬ šœŽ ī€· ( š“ āˆ˜ š‘“ ) āˆ’ 1 ī€ø = ī— ( šµ ) šµ āˆˆ ā„¬ ī— šœŽ ( šµ āˆ˜ š“ āˆ˜ š‘“ ) = šµ āˆˆ ā„¬ šœŽ ī€· š‘“ āˆ’ 1 ī€· š“ āˆ’ 1 ā‰„ ī— ( šµ ) ī€ø ī€ø šµ āˆˆ ā„¬ šœ ī€· š“ āˆ’ 1 ī€ø ( šµ ) = šœ ( šœ ) ( š“ ) , ( 4 . 5 ) we can prove the necessity.
Sufficiency. Suppose that š‘“ āˆ¶ ( š‘‹ , šœ ( šœŽ ) ) ā†’ ( š‘Œ , šœ ( šœ ) ) is ( š¼ , š¼ ) -fuzzy measurable. Then šœ ( šœŽ ) ( š‘“ ā† š¼ ( š“ ) ) ā‰„ šœ ( šœ ) ( š“ ) for any š“ āˆˆ š¼ š‘‹ . In particular, it follows that šœ ( šœŽ ) ( š‘“ ā† š¼ ( š“ ) ) ā‰„ šœ ( šœ ) ( š“ ) for any š“ āˆˆ šŸ š‘‹ . In order to prove that š‘“ āˆ¶ ( š‘‹ , šœŽ ) ā†’ ( š‘Œ , šœ ) is š¼ -fuzzifying measurable, we need to prove that šœŽ ( š‘“ āˆ’ 1 ( š“ ) ) ā‰„ šœ ( š“ ) for any š“ āˆˆ šŸ š‘‹ . In fact, for any š“ āˆˆ šŸ š‘‹ and for any šµ āˆˆ ā„¬ , if 0 , 1 āˆˆ šµ , then š“ āˆ’ 1 ( šµ ) = š‘‹ ; if 0 , 1 ĀØ ĀØ āˆˆ šµ , then š“ āˆ’ 1 ( šµ ) = āˆ… ; if only one of 0 and 1 is in šµ , then š“ āˆ’ 1 ( šµ ) = š“ or š“ āˆ’ 1 ( šµ ) = š“ ā€² . However, we have šœŽ ī€· š‘“ ā† š¼ ī€ø ī€· š‘“ ( š“ ) = šœŽ ā† š¼ ī€ø ī€· š‘“ ( š“ ) = šœŽ ā† š¼ ( ī€ø ī€· š‘“ š“ ) āˆ§ šœŽ ā† š¼ ( š“ ) ī…ž ī€ø = ī— šµ āˆˆ ā„¬ šœŽ ī‚€ ī€· š‘“ ā† šæ ī€ø ( š“ ) āˆ’ 1 ( ī‚ ī€· š‘“ šµ ) = šœ ( šœŽ ) ā† šæ ī€ø ī€· š“ ( š“ ) ā‰„ šœ ( šœ ) ( š“ ) = šœ ( šœ ) ( š“ ) āˆ§ šœ ( šœ ) ī…ž ī€ø = ī— šµ āˆˆ ā„¬ šœ ī€· š“ āˆ’ 1 ī€ø ( šµ ) = šœ ( š“ ) . ( 4 . 6 )
This shows that š‘“ āˆ¶ ( š‘‹ , šœŽ ) ā†’ ( š‘Œ , šœ ) is š¼ -fuzzifying measurable.

Corollary 4.4. Let ( š‘‹ , šœŽ ) and ( š‘Œ , šœ ) be two measurable spaces, and š‘“ āˆ¶ š‘‹ ā†’ š‘Œ is a mapping. Then š‘“ āˆ¶ ( š‘‹ , šœŽ ) ā†’ ( š‘Œ , šœ ) is measurable if and only if š‘“ āˆ¶ ( š‘‹ , šœ ( šœŽ ) ) ā†’ ( š‘Œ , šœ ( šœ ) ) is š¼ -measurable.

Acknowledgments

The author would like to thank the referees for their valuable comments and suggestions. The project is supported by the National Natural Science Foundation of China (10971242).