Abstract

The notion of ( ๐ฟ , ๐‘€ )-fuzzy ๐œŽ -algebras is introduced in the lattice value fuzzy set theory. It is a generalization of Klement's fuzzy ๐œŽ -algebras. In our definition of ( ๐ฟ , ๐‘€ )-fuzzy ๐œŽ -algebras, each ๐ฟ -fuzzy subset can be regarded as an ๐ฟ -measurable set to some degree.

1. Introduction and Preliminaries

In 1980, Klement established an axiomatic theory of fuzzy ๐œŽ -algebras in [1] in order to prepare a measure theory for fuzzy sets. In the definition of Klement's fuzzy ๐œŽ -algebra ( ๐‘‹ , ๐œŽ ) , ๐œŽ was defined as a crisp family of fuzzy subsets of a set ๐‘‹ satisfying certain set of axioms. In 1991, Biacino and Lettieri generalized Klement's fuzzy ๐œŽ -algebras to ๐ฟ -fuzzy setting [2].

In this paper, when both ๐ฟ and ๐‘€ are complete lattices, we define an ( ๐ฟ , ๐‘€ ) -fuzzy ๐œŽ -algebra on a nonempty set ๐‘‹ by means of a mapping ๐œŽ โˆถ ๐ฟ ๐‘‹ โ†’ ๐‘€ satisfying three axioms. Thus each ๐ฟ -fuzzy subset of ๐‘‹ can be regarded as an ๐ฟ -measurable set to some degree.

When ๐œŽ is an ( ๐ฟ , ๐‘€ ) -fuzzy ๐œŽ -algebra on ๐‘‹ , ( ๐‘‹ , ๐œŽ ) is called an ( ๐ฟ , ๐‘€ ) -fuzzy measurable space. An ( ๐ฟ , 2 ) -fuzzy ๐œŽ -algebra is also called an ๐ฟ - ๐œŽ -algebra. A Klement ๐œŽ -algebra can be viewed as a stratified [ 0 , 1 ] - ๐œŽ -algebra. A Biacino-Lettieri ๐ฟ - ๐œŽ -algebra can be viewed as a stratified ๐ฟ - ๐œŽ -algebra. A ( 2 , ๐‘€ ) -fuzzy ๐œŽ -algebra is also called an ๐‘€ -fuzzifying ๐œŽ -algebra. A crisp ๐œŽ -algebra can be regarded as a ( 2 , 2 ) -fuzzy ๐œŽ -algebra.

Throughout this paper, both ๐ฟ and ๐‘€ denote complete lattices, and ๐ฟ has an order-reversing involution โ€ฒ . ๐‘‹ is a nonempty set. ๐ฟ ๐‘‹ is the set of all ๐ฟ -fuzzy sets (or ๐ฟ -sets for short) on ๐‘‹ . We often do not distinguish a crisp subset ๐ด of ๐‘‹ and its character function ๐œ’ ๐ด . The smallest element and the largest element in ๐‘€ are denoted by โŸ‚ ๐‘€ and โŠค ๐‘€ , respectively.

The binary relation โ‰บ in ๐‘€ is defined as follows: for ๐‘Ž , ๐‘ โˆˆ ๐‘€ , ๐‘Ž โ‰บ ๐‘ if and only if for every subset ๐ท โІ ๐‘€ , the relation ๐‘ โฉฝ s u p ๐ท always implies the existence of ๐‘‘ โˆˆ ๐ท with ๐‘Ž โฉฝ ๐‘‘ [3]. { ๐‘Ž โˆˆ ๐‘€ โˆถ ๐‘Ž โ‰บ ๐‘ } is called the greatest minimal family of ๐‘ in the sense of [4], denoted by ๐›ฝ ( ๐‘ ) . Moreover, for ๐‘ โˆˆ ๐‘€ , we define ๐›ผ ( ๐‘ ) = { ๐‘Ž โˆˆ ๐‘€ โˆถ ๐‘Ž โ‰บ ๐‘œ ๐‘ ๐‘ } . In a completely distributive lattice ๐‘€ , there exist ๐›ผ ( ๐‘ ) and ๐›ฝ ( ๐‘ ) for each ๐‘ โˆˆ ๐‘€ , and โ‹ โ‹€ ๐‘ = ๐›ฝ ( ๐‘ ) = ๐›ผ ( ๐‘ ) (see [4]).

In [4], Wang thought that ๐›ฝ ( 0 ) = { 0 } and ๐›ผ ( 1 ) = { 1 } . In fact, it should be that ๐›ฝ ( 0 ) = โˆ… and ๐›ผ ( 1 ) = โˆ… .

For a complete lattice ๐ฟ , ๐ด โˆˆ ๐ฟ ๐‘‹ and ๐‘Ž โˆˆ ๐ฟ , we use the following notation:

๐ด [ ๐‘Ž ] = { ๐‘ฅ โˆˆ ๐‘‹ โˆถ ๐ด ( ๐‘ฅ ) โฉพ ๐‘Ž } . ( 1 . 1 )

If ๐ฟ is completely distributive, then we can define

๐ด [ ๐‘Ž ] = { ๐‘ฅ โˆˆ ๐‘‹ โˆถ ๐‘Ž ยจ ยจ โˆˆ ๐›ผ ( ๐ด ( ๐‘ฅ ) ) } . ( 1 . 2 )

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

Theorem 1.1 (see [4]). Let ๐‘€ be a completely distributive lattice and { ๐‘Ž ๐‘– โˆถ ๐‘– โˆˆ ฮฉ } โІ ๐‘€ . Then (1) โ‹€ ๐›ผ ( ๐‘– โˆˆ ฮฉ ๐‘Ž ๐‘– โ‹ƒ ) = ๐‘– โˆˆ ฮฉ ๐›ผ ( ๐‘Ž ๐‘– ) , that is, ๐›ผ is an โ‹€ โˆ’ โ‹ƒ map;(2) โ‹ ๐›ฝ ( ๐‘– โˆˆ ฮฉ ๐‘Ž ๐‘– โ‹ƒ ) = ๐‘– โˆˆ ฮฉ ๐›ฝ ( ๐‘Ž ๐‘– ) , that is, ๐›ฝ is a union-preserving map. For ๐‘Ž โˆˆ ๐ฟ and ๐ท โІ ๐‘‹ , we define two ๐ฟ -fuzzy sets ๐‘Ž โ‹€ ๐ท and ๐‘Ž โˆจ ๐ท as follows: ๎ƒฏ ( ๐‘Ž โˆง ๐ท ) ( ๐‘ฅ ) = ๐‘Ž , ๐‘ฅ โˆˆ ๐ท ; 0 , ๐‘ฅ ยจ ยจ ๎ƒฏ โˆˆ ๐ท . ( ๐‘Ž โˆจ ๐ท ) ( ๐‘ฅ ) = 1 , ๐‘ฅ โˆˆ ๐ท ; ๐‘Ž , ๐‘ฅ ยจ ยจ โˆˆ ๐ท . ( 1 . 3 ) Then for each ๐ฟ -fuzzy set ๐ด in ๐ฟ ๐‘‹ , it follows that ๎˜ ๐ด = ๐‘Ž โˆˆ ๐ฟ ๎€ท ๐‘Ž โˆง ๐ด [ ๐‘Ž ] ๎€ธ . ( 1 . 4 )

Theorem 1.2 (see [5, 7, 10]). If ๐ฟ is completely distributive, then for each ๐ฟ -fuzzy set ๐ด in ๐ฟ ๐‘‹ , we have (1) โ‹ ๐ด = ๐‘Ž โˆˆ ๐ฟ โ‹€ ๐ด ( ๐‘Ž [ ๐‘Ž ] โ‹€ ) = ๐‘Ž โˆˆ ๐ฟ ( ๐‘Ž โˆจ ๐ด [ ๐‘Ž ] ) ; (2) f o r a l l ๐‘Ž โˆˆ ๐ฟ , ๐ด [ ๐‘Ž ] = โ‹‚ ๐‘ โˆˆ ๐›ฝ ( ๐‘Ž ) ๐ด [ ๐‘ ] ; (3) f o r a l l ๐‘Ž โˆˆ ๐ฟ , ๐ด [ ๐‘Ž ] = โ‹‚ ๐‘Ž โˆˆ ๐›ผ ( ๐‘ ) ๐ด [ ๐‘ ] . For a family of ๐ฟ -fuzzy sets { ๐ด ๐‘– โˆถ ๐‘– โˆˆ ฮฉ } in ๐ฟ ๐‘‹ , it is easy to see that ๎ƒฉ ๎— ๐‘– โˆˆ ฮฉ ๐ด ๐‘– ๎ƒช [ ๐‘Ž ] = ๎™ ๐‘– โˆˆ ฮฉ ๎€ท ๐ด ๐‘– ๎€ธ [ ๐‘Ž ] . ( 1 . 5 ) If ๐ฟ is completely distributive, then it follows [7] that ๎ƒฉ ๎— ๐‘– โˆˆ ฮฉ ๐ด ๐‘– ๎ƒช [ ๐‘Ž ] = ๎™ ๐‘– โˆˆ ฮฉ ๎€ท ๐ด ๐‘– ๎€ธ [ ๐‘Ž ] . ( 1 . 6 )

Definition 1.3. Let ๐‘‹ be a nonempty set. A subset ๐œŽ of [ 0 , 1 ] ๐‘‹ is called a Klement fuzzy ๐œŽ -algebra if it satisfies the following three conditions:(1)for any constant fuzzy set ๐›ผ , ๐›ผ โˆˆ ๐œŽ ;(2)for any ๐ด โˆˆ [ 0 , 1 ] ๐‘‹ , 1 โˆ’ ๐ด โˆˆ ๐œŽ ;(3)for any { ๐ด ๐‘› โˆถ ๐‘› โˆˆ โ„• } โІ ๐œŽ , โ‹ ๐‘› โˆˆ โ„• ๐ด ๐‘› โˆˆ ๐œŽ .The fuzzy sets in ๐œŽ are called fuzzy measurable sets, and the pair ( ๐‘‹ , ๐œŽ ) a fuzzy measurable space.

Definition 1.4. Let ๐ฟ be a complete lattice with an order-reversing involution ๎…ž and ๐‘‹ a nonempty set. A subset ๐œŽ of ๐ฟ ๐‘‹ is called an ๐ฟ - ๐œŽ -algebra if it satisfies the following three conditions:(1)for any ๐‘Ž โˆˆ ๐ฟ , constant ๐ฟ -fuzzy set ๐‘Ž โ‹€ ๐œ’ ๐‘‹ โˆˆ ๐œŽ ;(2)for any ๐ด โˆˆ ๐ฟ ๐‘‹ , ๐ด ๎…ž โˆˆ ๐œŽ ;(3)for any { ๐ด ๐‘› โˆถ ๐‘› โˆˆ โ„• } โІ ๐œŽ , โ‹ ๐‘› โˆˆ โ„• ๐ด ๐‘› โˆˆ ๐œŽ .The ๐ฟ -fuzzy sets in ๐œŽ are called ๐ฟ -measurable sets, and the pair ( ๐‘‹ , ๐œŽ ) an ๐ฟ -measurable space.

2. ( ๐ฟ , ๐‘€ ) -Fuzzy ๐œŽ -Algebras

L. Biacino and A. Lettieri defined that an ๐ฟ - ๐œŽ -algebra ๐œŽ is a crisp subset of ๐ฟ ๐‘‹ . Now we consider an ๐‘€ -fuzzy subset ๐œŽ of ๐ฟ ๐‘‹ .

Definition. Let ๐‘‹ be a nonempty set. A mapping ๐œŽ โˆถ ๐ฟ ๐‘‹ โ†’ ๐‘€ is called an ( ๐ฟ , ๐‘€ ) -fuzzy ๐œŽ -algebra if it satisfies the following three conditions: (LMS1) ๐œŽ ( ๐œ’ โˆ… ) = โŠค ๐‘€ ;(LMS2) for any ๐ด โˆˆ ๐ฟ ๐‘‹ , ๐œŽ ( ๐ด ) = ๐œŽ ( ๐ด ๎…ž ) ;(LMS3) for any { ๐ด ๐‘› โˆถ ๐‘› โˆˆ โ„• } โІ ๐ฟ ๐‘‹ , โ‹ ๐œŽ ( ๐‘› โˆˆ โ„• ๐ด ๐‘› โ‹€ ) โ‰ฅ ๐‘› โˆˆ โ„• ๐œŽ ( ๐ด ๐‘› ) .An ( ๐ฟ , ๐‘€ ) -fuzzy ๐œŽ -algebra ๐œŽ is said to be stratified if and only if it satisfies the following condition:
(LMS1)* โ€‰โ€‰ โˆ€ ๐‘Ž โˆˆ ๐ฟ , โ‹€ ๐œ’ ๐œŽ ( ๐‘Ž ๐‘‹ ) = โŠค ๐‘€ .
If ๐œŽ is an ( ๐ฟ , ๐‘€ ) -fuzzy ๐œŽ -algebra, then ( ๐‘‹ , ๐œŽ ) is called an ( ๐ฟ , ๐‘€ ) -fuzzy measurable space.
An ( ๐ฟ , ๐Ÿ ) -fuzzy ๐œŽ -algebra is also called an ๐ฟ - ๐œŽ -algebra, and an ( ๐ฟ , ๐Ÿ ) -fuzzy measurable space is also called an ๐ฟ -measurable space.
A ( ๐Ÿ , ๐‘€ ) -fuzzy ๐œŽ -algebra is also called an ๐‘€ -fuzzifying ๐œŽ -algebra, and a ( ๐Ÿ , ๐‘€ ) -fuzzy measurable space is also called an ๐‘€ -fuzzifying measurable space.
Obviously a crisp measurable space can be regarded as a ( ๐Ÿ , ๐Ÿ ) -fuzzy measurable space.
If ๐œŽ is an ( ๐ฟ , ๐‘€ ) -fuzzy ๐œŽ -algebra, then ๐œŽ ( ๐ด ) can be regarded as the degree to which ๐ด is an ๐ฟ -measurable set.

Remark 2.2. If a subset ๐œŽ of ๐ฟ ๐‘‹ is regarded as a mapping ๐œŽ โˆถ ๐ฟ ๐‘‹ โ†’ ๐Ÿ , then ๐œŽ is an ๐ฟ - ๐œŽ -algebra if and only if it satisfies the following conditions:(LS1) ๐œ’ โˆ… โˆˆ ๐œŽ ;(LS2) ๐ด โˆˆ ๐œŽ โ‡’ ๐ด ๎…ž โˆˆ ๐œŽ ;(LS3) for any { ๐ด ๐‘› โˆถ ๐‘› โˆˆ โ„• } โІ ๐œŽ , โ‹ ๐‘› โˆˆ โ„• ๐ด ๐‘› โˆˆ ๐œŽ . Thus we easily see that a Klement ๐œŽ -algebra is exactly a stratified [ 0 , 1 ] - ๐œŽ -algebra, and a Biacino-Lettieri ๐ฟ - ๐œŽ -algebra is exactly a stratified ๐ฟ - ๐œŽ -algebra.
Moreover, when ๐ฟ = ๐Ÿ , a mapping ๐œŽ โˆถ ๐Ÿ ๐‘‹ โ†’ ๐‘€ is an ๐‘€ -fuzzifying ๐œŽ -algebra if and only if it satisfies the following conditions:
(MS1) ๐œŽ ( โˆ… ) = โŠค ๐‘€ ;(MS2) for any ๐ด โˆˆ ๐Ÿ ๐‘‹ , ๐œŽ ( ๐ด ) = ๐œŽ ( ๐ด ๎…ž ) ;(MS3) for any { ๐ด ๐‘› โˆถ ๐‘› โˆˆ โ„• } โІ 2 ๐‘‹ , โ‹ ๐œŽ ( ๐‘› โˆˆ โ„• ๐ด ๐‘› โ‹€ ) โ‰ฅ ๐‘› โˆˆ โ„• ๐œŽ ( ๐ด ๐‘› ) .

Example 2.3. Letโ€‰โ€‰ ( ๐‘‹ , ๐œŽ ) be a crisp measurable space. Define ๐œ’ ๐œŽ โˆถ ๐Ÿ ๐‘‹ โ†’ [ 0 , 1 ] by ๐œ’ ๐œŽ ๎ƒฏ ( ๐ด ) = 1 , ๐ด โˆˆ ๐œŽ ; 0 , ๐ด ยจ ยจ โˆˆ ๐œŽ . ( 2 . 1 ) Then it is easy to prove that ( ๐‘‹ , ๐œ’ ๐œŽ ) is a [ 0 , 1 ] -fuzzifying measurable space.

Example 2.4. Let ๐‘‹ be a nonempty set and ๐œŽ โˆถ ๐Ÿ ๐‘‹ โ†’ [ 0 , 1 ] a mapping defined by ๎ƒฏ ๐œŽ ( ๐ด ) = 1 , ๐ด โˆˆ { โˆ… , ๐‘‹ } ; 0 . 5 , ๐ด ยจ ยจ โˆˆ { โˆ… , ๐‘‹ } . ( 2 . 2 ) Then it is easy to prove that ( ๐‘‹ , ๐œŽ ) is a [ 0 , 1 ] -fuzzifying measurable space. If ๐ด โˆˆ ๐Ÿ ๐‘‹ with ๐ด ยจ ยจ โˆˆ { โˆ… , ๐‘‹ } , then 0.5 is the degree to which ๐ด is measurable.

Example 2.5. Let ๐‘‹ be a nonempty set and ๐œŽ โˆถ [ 0 , 1 ] ๐‘‹ โ†’ [ 0 , 1 ] a mapping defined by ๎ƒฏ ๎€ฝ ๐œ’ ๐œŽ ( ๐ด ) = 1 , ๐ด โˆˆ โˆ… , ๐œ’ ๐‘‹ ๎€พ ; 0 . 5 , ๐ด ยจ ยจ โˆˆ ๎€ฝ ๐œ’ โˆ… , ๐œ’ ๐‘‹ ๎€พ . ( 2 . 3 ) Then it is easy to prove that ( ๐‘‹ , ๐œŽ ) is a ( [ 0 , 1 ] , [ 0 , 1 ] ) -fuzzy measurable space. If ๐ด โˆˆ [ 0 , 1 ] ๐‘‹ with ๐ด ยจ ยจ โˆˆ { ๐œ’ โˆ… , ๐œ’ ๐‘‹ } , then 0.5 is the degree to which ๐ด is [ 0 , 1 ] -measurable.

Proposition 2.6. Let ( ๐‘‹ , ๐œŽ ) be an ( ๐ฟ , ๐‘€ ) -fuzzy measurable spaces. Then for any { ๐ด ๐‘› โˆถ ๐‘› โˆˆ โ„• } โІ ๐ฟ ๐‘‹ , โ‹€ ๐œŽ ( ๐‘› โˆˆ โ„• ๐ด ๐‘› โ‹€ ) โ‰ฅ ๐‘› โˆˆ โ„• ๐œŽ ( ๐ด ๐‘› ) .

Proof. This can be proved from the following fact: ๐œŽ ๎ƒฉ ๎— ๐‘› โˆˆ โ„• ๐ด ๐‘› ๎ƒช ๎ƒฉ ๎˜ = ๐œŽ ๐‘› โˆˆ โ„• ๎€ท ๐ด ๐‘› ๎€ธ ๎…ž ๎ƒช โ‰ฅ ๎— ๐‘› โˆˆ โ„• ๐œŽ ๎‚€ ๎€ท ๐ด ๐‘› ๎€ธ ๎…ž ๎‚ = ๎— ๐‘› โˆˆ โ„• ๐œŽ ๎€ท ๐ด ๐‘› ๎€ธ . ( 2 . 4 ) The next two theorems give characterizations of an ( ๐ฟ , ๐‘€ ) -fuzzy ๐œŽ -algebra.

Theorem 2.7. A mapping ๐œŽ โˆถ ๐ฟ ๐‘‹ โ†’ ๐‘€ is an ( ๐ฟ , ๐‘€ ) -fuzzy ๐œŽ -algebra if and only if for each ๐‘Ž โˆˆ ๐‘€ โงต { โŸ‚ ๐‘€ } , ๐œŽ [ ๐‘Ž ] is an ๐ฟ - ๐œŽ -algebra.

Proof. The proof is obvious and is omitted.

Corollary 2.8. A mapping ๐œŽ โˆถ ๐Ÿ ๐‘‹ โ†’ ๐‘€ is an ๐‘€ -fuzzifying ๐œŽ -algebra if and only if for each ๐‘Ž โˆˆ ๐‘€ โงต { โŸ‚ ๐‘€ } , ๐œŽ [ ๐‘Ž ] is a ๐œŽ -algebra.

Theorem 2.9. If ๐‘€ is completely distributive, then a mapping ๐œŽ โˆถ ๐ฟ ๐‘‹ โ†’ ๐‘€ is an ( ๐ฟ , ๐‘€ ) -fuzzy ๐œŽ -algebra if and only if for each ๐‘Ž โˆˆ ๐›ผ ( โŸ‚ ๐‘€ ) , ๐œŽ [ ๐‘Ž ] is an ๐ฟ - ๐œŽ -algebra.

Proof. Necessity.. Suppose that ๐œŽ โˆถ ๐ฟ ๐‘‹ โ†’ ๐‘€ is an ( ๐ฟ , ๐‘€ ) -fuzzy ๐œŽ -algebra and ๐‘Ž โˆˆ ๐›ผ ( โŸ‚ ๐‘€ ) . Now we prove that ๐œŽ [ ๐‘Ž ] is an ๐ฟ - ๐œŽ -algebra.(LS1) By ๐œŽ ( ๐œ’ โˆ… ) = โŠค ๐‘€ and ๐›ผ ( โŠค ๐‘€ ) = โˆ… , we know that ๐‘Ž ยจ ยจ โˆˆ ๐›ผ ( ๐œŽ ( ๐œ’ โˆ… ) ) ; this implies that ๐œ’ โˆ… โˆˆ ๐œŽ [ ๐‘Ž ] .(LS2) If ๐ด โˆˆ ๐œŽ [ ๐‘Ž ] , then ๐‘Ž ยจ ยจ โˆˆ ๐›ผ ( ๐œŽ ( ๐ด ) ) = ๐›ผ ( ๐œŽ ( ๐ด ๎…ž ) ) ; this shows that ๐ด ๎…ž โˆˆ ๐œŽ [ ๐‘Ž ] .(LS3) If { ๐ด ๐‘– โˆถ ๐‘– โˆˆ ฮฉ } โІ ๐œŽ [ ๐‘Ž ] , then f o r a l l ๐‘– โˆˆ ฮฉ , ๐‘Ž ยจ ยจ โˆˆ ๐›ผ ( ๐œŽ ( ๐ด ๐‘– ) ) . Hence ๐‘Ž ยจ ยจ โˆˆ โ‹ƒ ๐‘– โˆˆ ฮฉ ๐›ผ ( ๐œŽ ( ๐ด ๐‘– ) ) . By โ‹ ๐œŽ ( ๐‘– โˆˆ ฮฉ ๐ด ๐‘– โ‹€ ) โ‰ฅ ๐‘– โˆˆ ฮฉ ๐œŽ ( ๐ด ๐‘– ) , we know that ๐›ผ ๎ƒฉ ๐œŽ ๎ƒฉ ๎˜ ๐‘– โˆˆ ฮฉ ๐ด ๐‘– ๎ƒฉ ๎— ๎ƒช ๎ƒช โІ ๐›ผ ๐‘– โˆˆ ฮฉ ๐œŽ ๎€ท ๐ด ๐‘– ๎€ธ ๎ƒช = ๎š ๐‘– โˆˆ ฮฉ ๐›ผ ๎€ท ๐œŽ ๎€ท ๐ด ๐‘– . ๎€ธ ๎€ธ ( 2 . 5 ) This shows that ๐‘Ž ยจ ยจ โ‹ โˆˆ ๐›ผ ( ๐œŽ ( ๐‘– โˆˆ ฮฉ ๐ด ๐‘– ) ) . Therefore, โ‹ ๐‘– โˆˆ ฮฉ ๐ด ๐‘– โˆˆ ๐œŽ [ ๐‘Ž ] . The proof is completed.

Corollary 2.10. If ๐‘€ is completely distributive, then a mapping ๐œŽ โˆถ ๐Ÿ ๐‘‹ โ†’ ๐‘€ is an ๐‘€ -fuzzifying ๐œŽ -algebra if and only if for each ๐‘Ž โˆˆ ๐›ผ ( โŸ‚ ๐‘€ ) , ๐œŽ [ ๐‘Ž ] is a ๐œŽ -algebra.

Now we consider the conditions that a family of ๐ฟ - ๐œŽ -algebras forms an ( ๐ฟ , ๐‘€ ) -fuzzy ๐œŽ -algebra. By Theorem 1.2, we can obtain the following result.

Corollary 2.11. If ๐‘€ is completely distributive, and ๐œŽ is an ( ๐ฟ , ๐‘€ ) -fuzzy ๐œŽ -algebra, then (1) ๐œŽ [ ๐‘ ] โІ ๐œŽ [ ๐‘Ž ] for any ๐‘Ž , ๐‘ โˆˆ ๐‘€ โงต { โŸ‚ ๐‘€ } with ๐‘Ž โˆˆ ๐›ฝ ( ๐‘ ) ;(2) ๐œŽ [ ๐‘ ] โІ ๐œŽ [ ๐‘Ž ] for any ๐‘Ž , ๐‘ โˆˆ ๐›ผ ( โŸ‚ ๐‘€ ) with ๐‘ โˆˆ ๐›ผ ( ๐‘Ž ) .

Theorem 2.12. Let ๐‘€ be completely distributive, and let { ๐œŽ ๐‘Ž โˆถ ๐‘Ž โˆˆ ๐›ผ ( โŸ‚ ๐‘€ ) } be a family of ๐ฟ - ๐œŽ -algebras. If ๐œŽ ๐‘Ž = โ‹‚ { ๐œŽ ๐‘ โˆถ ๐‘Ž โˆˆ ๐›ผ ( ๐‘ ) } for all ๐‘Ž โˆˆ ๐›ผ ( โŸ‚ ๐‘€ ) , then there exists an ( ๐ฟ , ๐‘€ ) -fuzzy ๐œŽ -algebra ๐œŽ such that ๐œŽ [ ๐‘Ž ] = ๐œŽ ๐‘Ž .

Proof. Suppose that ๐œŽ ๐‘Ž = โ‹‚ { ๐œŽ ๐‘ โˆถ ๐‘Ž โˆˆ ๐›ผ ( ๐‘ ) } for all ๐‘Ž โˆˆ ๐›ผ ( โŸ‚ ๐‘€ ) . Define ๐œŽ โˆถ ๐ฟ ๐‘‹ โ†’ ๐‘€ by ๎— ๐œŽ ( ๐ด ) = ๐‘Ž โˆˆ ๐‘€ ( ๐‘Ž โˆจ ๐œŽ ๐‘Ž ๎— ( ๐ด ) ) = { ๐‘Ž โˆˆ ๐‘€ โˆถ ๐ด ยจ ยจ โˆˆ ๐œŽ ๐‘Ž } . ( 2 . 6 ) By Theorem 1.2, we can obtain that ๐œŽ [ ๐‘Ž ] = ๐œŽ ๐‘Ž .

Corollary 2.13. Let ๐‘€ be completely distributive, and let { ๐œŽ ๐‘Ž โˆถ ๐‘Ž โˆˆ ๐›ผ ( โŸ‚ ๐‘€ ) } be a family of ๐œŽ -algebras. If ๐œŽ ๐‘Ž = โ‹‚ { ๐œŽ ๐‘ โˆถ ๐‘Ž โˆˆ ๐›ผ ( ๐‘ ) } for all ๐‘Ž โˆˆ ๐›ผ ( โŸ‚ ๐‘€ ) , then there exists an ๐‘€ -fuzzifying ๐œŽ -algebra ๐œŽ such that ๐œŽ [ ๐‘Ž ] = ๐œŽ ๐‘Ž .

Theorem 2.14. Let ๐‘€ be completely distributive, and let { ๐œŽ ๐‘Ž โˆถ ๐‘Ž โˆˆ ๐‘€ โงต { โŸ‚ ๐‘€ } } be a family of ๐ฟ - ๐œŽ -algebra. If ๐œŽ ๐‘Ž = โ‹‚ { ๐œŽ ๐‘ โˆถ ๐‘ โˆˆ ๐›ฝ ( ๐‘Ž ) } for all ๐‘Ž โˆˆ ๐‘€ โงต { โŸ‚ ๐‘€ } , then there exists an ( ๐ฟ , ๐‘€ ) -fuzzy ๐œŽ -algebra ๐œŽ such that ๐œŽ [ ๐‘Ž ] = ๐œŽ ๐‘Ž .

Proof. Suppose that ๐œŽ ๐‘Ž = โ‹‚ { ๐œŽ ๐‘ โˆถ ๐‘ โˆˆ ๐›ฝ ( ๐‘Ž ) } for all ๐‘Ž โˆˆ ๐‘€ โงต { โŸ‚ ๐‘€ } . Define ๐œŽ โˆถ ๐ฟ ๐‘‹ โ†’ ๐‘€ by ๎˜ ๐œŽ ( ๐ด ) = ๐‘Ž โˆˆ ๐‘€ ๎€ท ๐‘Ž โˆง ๐œŽ ๐‘Ž ๎€ธ = ๎˜ ๎€ฝ ( ๐ด ) ๐‘Ž โˆˆ ๐‘€ โˆถ ๐ด โˆˆ ๐œŽ ๐‘Ž ๎€พ . ( 2 . 7 ) By Theorem 1.2, we can obtain ๐œŽ [ ๐‘Ž ] = ๐œŽ ๐‘Ž .

Corollary 2.15. Let ๐‘€ be completely distributive, and let { ๐œŽ ๐‘Ž โˆถ ๐‘Ž โˆˆ ๐‘€ โงต { โŸ‚ ๐‘€ } } be a family of ๐œŽ -algebra. If ๐œŽ ๐‘Ž = โ‹‚ { ๐œŽ ๐‘ โˆถ ๐‘ โˆˆ ๐›ฝ ( ๐‘Ž ) } for all ๐‘Ž โˆˆ ๐‘€ โงต { โŸ‚ ๐‘€ } , then there exists an ๐‘€ -fuzzifying ๐œŽ -algebra ๐œŽ such that ๐œŽ [ ๐‘Ž ] = ๐œŽ ๐‘Ž .

Theorem 2.16. Let { ๐œŽ ๐‘– โˆถ ๐‘– โˆˆ ฮฉ } be a family of ( ๐ฟ , ๐‘€ ) -fuzzy ๐œŽ -algebra on ๐‘‹ . Then โ‹€ ๐‘– โˆˆ ฮฉ ๐œŽ ๐‘– is an ( ๐ฟ , ๐‘€ ) -fuzzy ๐œŽ -algebra on ๐‘‹ , where โ‹€ ๐‘– โˆˆ ฮฉ ๐œŽ ๐‘– โˆถ ๐ฟ ๐‘‹ โ†’ ๐‘€ is defined by ( โ‹€ ๐‘– โˆˆ ฮฉ ๐œŽ ๐‘– โ‹€ ) ( ๐ด ) = ๐‘– โˆˆ ฮฉ ๐œŽ ๐‘– ( ๐ด ) .

Proof. This is straightforward.

3. ( ๐ฟ , ๐‘€ ) -Fuzzy Measurable Functions

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

Theorem 3.1. Let ( ๐‘Œ , ๐œ ) be an ( ๐ฟ , ๐‘€ ) -fuzzy measurable space and ๐‘“ โˆถ ๐‘‹ โ†’ ๐‘Œ a mapping. Define a mapping ๐‘“ โ† ๐ฟ ( ๐œ ) โˆถ ๐ฟ ๐‘‹ โ†’ ๐‘€ by f o r a l l ๐ด โˆˆ ๐ฟ ๐‘‹ , ๐‘“ โ† ๐ฟ ๎˜ ๎€ฝ ( ๐œ ) ( ๐ด ) = ๐œ ( ๐ต ) โˆถ ๐‘“ โ† ๐ฟ ๎€พ , ( ๐ต ) = ๐ด w h e r e โˆ€ ๐‘ฅ โˆˆ ๐‘‹ , ๐‘“ โ† ๐ฟ ( ๐ต ) ( ๐‘ฅ ) = ๐ต ( ๐‘“ ( ๐‘ฅ ) ) . ( 3 . 1 ) Then ( ๐‘‹ , ๐‘“ โ† ๐ฟ ( ๐œ ) ) is an ( ๐ฟ , ๐‘€ ) -fuzzy measurable space.

Proof. (LMS1) holds from the following equality: ๐‘“ โ† ๐ฟ ๎€ท ๐œ’ ( ๐œ ) โˆ… ๎€ธ = ๎˜ ๎€ฝ ๐œ ( ๐ต ) โˆถ ๐‘“ โ† ๐ฟ ( ๐ต ) = ๐œ’ โˆ… ๎€พ ๎€ท ๐œ’ = ๐œ โˆ… ๎€ธ = โŠค ๐‘€ . ( 3 . 2 )
(LMS2) can be shown from the following fact: f o r a l l ๐ด โˆˆ ๐ฟ ๐‘‹ , ๐‘“ โ† ๐ฟ ๎˜ ๎€ฝ ( ๐œ ) ( ๐ด ) = ๐œ ( ๐ต ) โˆถ ๐‘“ โ† ๐ฟ ๎€พ = ๎˜ ๎€ฝ ๐œ ๎€ท ๐ต ( ๐ต ) = ๐ด ๎…ž ๎€ธ โˆถ ๐‘“ โ† ๐ฟ ๎€ท ๐ต ๎…ž ๎€ธ = ๐‘“ โ† ๐ฟ ( ๐ต ) ๎…ž = ๐ด ๎…ž ๎€พ = ๐‘“ โ† ๐ฟ ๎€ท ๐ด ( ๐œ ) ๎…ž ๎€ธ . ( 3 . 3 )
(LMS3) for any { ๐ด ๐‘› โˆถ ๐‘› โˆˆ โ„• } โІ ๐ฟ ๐‘‹ , by ๐‘“ โ† ๐ฟ ๎ƒฉ ๎˜ ( ๐œ ) ๐‘› โˆˆ โ„• ๐ด ๐‘› ๎ƒช = ๎˜ ๎ƒฏ ๐œ ( ๐ต ) โˆถ ๐‘“ โ† ๐ฟ ๎˜ ( ๐ต ) = ๐‘› โˆˆ โ„• ๐ด ๐‘› ๎ƒฐ โ‰ฅ ๎˜ ๎ƒฏ ๐œ ๎ƒฉ ๎˜ ๐‘› โˆˆ โ„• ๐ต ๐‘› ๎ƒช โˆถ ๐‘“ โ† ๐ฟ ๎€ท ๐ต ๐‘› ๎€ธ = ๐ด ๐‘› ๎ƒฐ โ‰ฅ ๎— ๐‘› โˆˆ โ„• ๐‘“ โ† ๐ฟ ( ๎€ท ๐ด ๐œ ) ๐‘› ๎€ธ ( 3 . 4 ) w e c a n p r o v e ( L M S 3 ) .

Definition 3.2. Let ( ๐‘‹ , ๐œŽ ) and ( ๐‘Œ , ๐œ ) be ( ๐ฟ , ๐‘€ ) -fuzzy measurable spaces. A mapping ๐‘“ โˆถ ๐‘‹ โ†’ ๐‘Œ is called ( ๐ฟ , ๐‘€ ) -fuzzy measurable if ๐œŽ ( ๐‘“ โ† ๐ฟ ( ๐ต ) ) โ‰ฅ ๐œ ( ๐ต ) for all ๐ต โˆˆ ๐ฟ ๐‘Œ .
An ( ๐ฟ , ๐Ÿ ) -fuzzy measurable mapping is called an ๐ฟ -measurable mapping, and a ( ๐Ÿ , ๐‘€ ) -fuzzy measurable mapping is called an ๐‘€ -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 ( ๐ฟ , ๐‘€ ) -fuzzy measurable mappings.

Theorem 3.3. Let ( ๐‘‹ , ๐œŽ ) and ( ๐‘Œ , ๐œ ) be two ( ๐ฟ , ๐‘€ ) -fuzzy measurable spaces. A mapping ๐‘“ โˆถ ๐‘‹ โ†’ ๐‘Œ is ( ๐ฟ , ๐‘€ ) -fuzzy measurable if and only if ๐‘“ โ† ๐ฟ ( ๐œ ) ( ๐ด ) โ‰ค ๐œŽ ( ๐ด ) for all ๐ด โˆˆ ๐ฟ ๐‘‹ .

Proof. Necessity.. If ๐‘“ โˆถ ๐‘‹ โ†’ ๐‘Œ is ( ๐ฟ , ๐‘€ ) -fuzzy measurable, then ๐œŽ ( ๐‘“ โ† ๐ฟ ( ๐ต ) ) โ‰ฅ ๐œ ( ๐ต ) for all ๐ต โˆˆ ๐ฟ ๐‘Œ . Hence for all ๐ต โˆˆ ๐ฟ ๐‘Œ , we have ๐‘“ โ† ๐ฟ ๎˜ ๎€ฝ ( ๐œ ) ( ๐ด ) = ๐œ ( ๐ต ) โˆถ ๐‘“ โ† ๐ฟ ๎€พ โ‰ค ๎˜ ๎€ฝ ๐œŽ ๎€ท ๐‘“ ( ๐ต ) = ๐ด โ† ๐ฟ ๎€ธ ( ๐ต ) โˆถ ๐‘“ โ† ๐ฟ ๎€พ ( ๐ต ) = ๐ด = ๐œŽ ( ๐ด ) . ( 3 . 5 ) Sufficiency.. If ๐‘“ โ† ๐ฟ ( ๐œ ) ( ๐ด ) โ‰ค ๐œŽ ( ๐ด ) for all ๐ด โˆˆ ๐ฟ ๐‘‹ , then ๐œ ( ๐ต ) โ‰ค ๐‘“ โ† ๐ฟ ( ๐œ ) ( ๐‘“ โ† ๐ฟ ( ๐ต ) ) โ‰ค ๐œŽ ( ๐‘“ โ† ๐ฟ ( ๐ต ) ) for all ๐ต โˆˆ ๐ฟ ๐‘Œ ; this shows that ๐‘“ โˆถ ๐‘‹ โ†’ ๐‘Œ is ( ๐ฟ , ๐‘€ ) -fuzzy measurable.

The next three theorems are trivial.

Theorem 3.4. If ๐‘“ โˆถ ( ๐‘‹ , ๐œŽ ) โ†’ ( ๐‘Œ , ๐œ ) and ๐‘“ โˆถ ( ๐‘Œ , ๐œ ) โ†’ ( ๐‘ , ๐œŒ ) are ( ๐ฟ , ๐‘€ ) -fuzzy measurable, then ๐‘” โˆ˜ ๐‘“ โˆถ ( ๐‘‹ , ๐œŽ ) โ†’ ( ๐‘ , ๐œŒ ) is ( ๐ฟ , ๐‘€ ) -fuzzy measurable.

Theorem 3.5. Let ( ๐‘‹ , ๐œŽ ) and ( ๐‘Œ , ๐œ ) be ( ๐ฟ , ๐‘€ ) -fuzzy measurable spaces. Then a mapping ๐‘“ โˆถ ( ๐‘‹ , ๐œŽ ) โ†’ ( ๐‘Œ , ๐œ ) is ( ๐ฟ , ๐‘€ ) -fuzzy measurable if and only if ๐‘“ โˆถ ( ๐‘‹ , ๐œŽ [ ๐‘Ž ] ) โ†’ ( ๐‘Œ , ๐œ [ ๐‘Ž ] ) is ๐ฟ -measurable for any ๐‘Ž โˆˆ ๐‘€ โงต { โŸ‚ ๐‘€ } .

Theorem 3.6. Let ๐‘€ be completely distributive, and let ( ๐‘‹ , ๐œŽ ) and ( ๐‘Œ , ๐œ ) be ( ๐ฟ , ๐‘€ ) -fuzzy measurable spaces. Then a mapping ๐‘“ โˆถ ( ๐‘‹ , ๐œŽ ) โ†’ ( ๐‘Œ , ๐œ ) is ( ๐ฟ , ๐‘€ ) -fuzzy measurable if and only if ๐‘“ โˆถ ( ๐‘‹ , ๐œŽ [ ๐‘Ž ] ) โ†’ ( ๐‘Œ , ๐œ [ ๐‘Ž ] ) is ๐ฟ -measurable for any ๐‘Ž โˆˆ ๐›ผ ( โŸ‚ ๐‘€ ) .

Corollary 3.7. Let ( ๐‘‹ , ๐œŽ ) and ( ๐‘Œ , ๐œ ) be ๐‘€ -fuzzifying measurable spaces. Then a mapping ๐‘“ โˆถ ( ๐‘‹ , ๐œŽ ) โ†’ ( ๐‘Œ , ๐œ ) is ๐‘€ -fuzzifying measurable if and only if ๐‘“ โˆถ ( ๐‘‹ , ๐œŽ [ ๐‘Ž ] ) โ†’ ( ๐‘Œ , ๐œ [ ๐‘Ž ] ) is measurable for any ๐‘Ž โˆˆ ๐‘€ โงต { โŸ‚ ๐‘€ } .

Corollary 3.8. Let ๐‘€ be completely distributive, and let ( ๐‘‹ , ๐œŽ ) and ( ๐‘Œ , ๐œ ) be ๐‘€ -fuzzifying measurable spaces. Then a mapping ๐‘“ โˆถ ( ๐‘‹ , ๐œŽ ) โ†’ ( ๐‘Œ , ๐œ ) is ๐‘€ -fuzzifying measurable if and only if ๐‘“ โˆถ ( ๐‘‹ , ๐œŽ [ ๐‘Ž ] ) โ†’ ( ๐‘Œ , ๐œ [ ๐‘Ž ] ) is measurable for any ๐‘Ž โˆˆ ๐›ผ ( โŸ‚ ๐‘€ ) .

4. ( ๐ผ , ๐ผ ) -Fuzzy ๐œŽ -Algebras Generated by ๐ผ -Fuzzifying ๐œŽ -Algebras

In this section, โ„ฌ will be used to denote the ๐œŽ -algebra of Borel subsets of ๐ผ = [ 0 , 1 ] .

Theorem 4.1. Let ( ๐‘‹ , ๐œŽ ) be an ๐ผ -fuzzifying measurable space. Define a mapping ๐œ ( ๐œŽ ) โˆถ ๐ผ ๐‘‹ โ†’ ๐ผ by ๎— ๐œ ( ๐œŽ ) ( ๐ด ) = ๐ต โˆˆ โ„ฌ ๐œŽ ๎€ท ๐ด โˆ’ 1 ๎€ธ . ( ๐ต ) ( 4 . 1 ) Then ๐œ ( ๐œŽ ) is a stratified ( ๐ผ , ๐ผ ) -fuzzy ๐œŽ -algebra, which is said to be the ( ๐ผ , ๐ผ ) -fuzzy ๐œŽ -algebra generated by ๐œŽ .

Proof. (LMS1) For any ๐ต โˆˆ โ„ฌ and for any ๐‘Ž โˆˆ ๐ผ , if ๐‘Ž โˆˆ ๐ต , then โ‹€ ๐œ’ ( ๐‘Ž ๐‘‹ ) โˆ’ 1 ( ๐ต ) = ๐‘‹ ; if ๐‘Ž ยจ ยจ โˆˆ ๐ต , then โ‹€ ๐œ’ ( ๐‘Ž ๐‘‹ ) โˆ’ 1 ( ๐ต ) = โˆ… . However, we have that โ‹€ ๐œ’ ๐œŽ ( ( ๐‘Ž ๐‘‹ ) โˆ’ 1 ( ๐ต ) ) = 1 . This shows that โ‹€ ๐œ’ ๐œ ( ๐œŽ ) ( ๐‘Ž ๐‘‹ ) = 1 .
(LMS2) f o r a l l ๐ด โˆˆ ๐ผ ๐‘‹ and f o r a l l ๐ต โˆˆ โ„ฌ , we have ๎€ท ๐ด ๐œ ( ๐œŽ ) ๎…ž ๎€ธ = ๎— ๐ต โˆˆ โ„ฌ ๐œŽ ๎€ท ( 1 โˆ’ ๐ด ) โˆ’ 1 ๎€ธ = ๎— ( ๐ต ) ๐ต โˆˆ โ„ฌ ๐œŽ = ๎— ( { ๐‘ฅ โˆˆ ๐‘‹ โˆถ 1 โˆ’ ๐ด ( ๐‘ฅ ) โˆˆ ๐ต } ) ๐ต โˆˆ โ„ฌ ๐œŽ ( { ๐‘ฅ โˆˆ ๐‘‹ โˆถ โˆƒ ๐‘ โˆˆ ๐ต , s . t = ๎— . ๐ด ( ๐‘ฅ ) = 1 โˆ’ ๐‘ } ) ๐ต โˆˆ โ„ฌ ๐œŽ ๎€ท ๐ด โˆ’ 1 ๎€ธ ( ๐ต ) = ๐œ ( ๐œŽ ) ( ๐ด ) . ( 4 . 2 )
(LMS3) for any { ๐ด ๐‘› โˆถ ๐‘› โˆˆ โ„• } โІ ๐ฟ ๐‘‹ and f o r a l l ๐ต โˆˆ โ„ฌ , by ๐œ ๎ƒฉ ๎˜ ( ๐œŽ ) ๐‘› โˆˆ โ„• ๐ด ๐‘› ๎ƒช = ๎— ๐ต โˆˆ โ„ฌ ๐œŽ โŽ› โŽœ โŽœ โŽ ๎ƒฉ ๎˜ ๐‘› โˆˆ โ„• ๐ด ๐‘› ๎ƒช โˆ’ 1 โŽž โŽŸ โŽŸ โŽ  = ๎— ( ๐ต ) ๐ต โˆˆ โ„ฌ ๐œŽ ๎ƒฉ ๎š ๐‘› โˆˆ โ„• ๐ด ๐‘› โˆ’ 1 ๎ƒช โ‰ฅ ๎— ( ๐ต ) ๐ต โˆˆ โ„ฌ ๎— ๐‘› โˆˆ โ„• ๐œŽ ๎€ท ๐ด ๐‘› โˆ’ 1 ๎€ธ = ๎— ( ๐ต ) ๐‘› โˆˆ โ„• ๎— ๐ต โˆˆ โ„ฌ ๐œŽ ๎€ท ๐ด ๐‘› โˆ’ 1 ๎€ธ = ๎— ( ๐ต ) ๐‘› โˆˆ โ„• ๎€ท ๐ด ๐œ ( ๐œŽ ) ๐‘› ๎€ธ , ( 4 . 3 )
we obtain โ‹ ๐œ ( ๐œŽ ) ( ๐‘› โˆˆ โ„• ๐ด ๐‘› โ‹€ ) โ‰ฅ ๐‘› โˆˆ โ„• ๐œ ( ๐œŽ ) ( ๐ด ๐‘› ) .

Corollary 4.2. Let ( ๐‘‹ , ๐œŽ ) be a measurable space. Define a subset ๐œ ( ๐œŽ ) โІ ๐ผ ๐‘‹ ( can be viewed as a mapping ๐œ ( ๐œŽ ) โˆถ ๐ผ ๐‘‹ โ†’ ๐Ÿ ) by ๎€ฝ ๐œ ( ๐œŽ ) = ๐ด โˆˆ ๐ผ ๐‘‹ โˆถ โˆ€ ๐ต โˆˆ โ„ฌ , ๐ด โˆ’ 1 ๎€พ . ( ๐ต ) โˆˆ ๐œŽ ( 4 . 4 ) Then ๐œ ( ๐œŽ ) is a stratified ๐ผ - ๐œŽ -algebra.

From Corollary 4.2, we see that the functor ๐œ in Theorem 4.1 is a generalization of Klement functor ๐œ .

Theorem 4.3. Let ( ๐‘‹ , ๐œŽ ) and ( ๐‘Œ , ๐œ ) be two ๐ผ -fuzzifying measurable spaces, and ๐‘“ โˆถ ๐‘‹ โ†’ ๐‘Œ is a map. Then ๐‘“ โˆถ ( ๐‘‹ , ๐œŽ ) โ†’ ( ๐‘Œ , ๐œ ) is ๐ผ -fuzzifying measurable if and only if ๐‘“ โˆถ ( ๐‘‹ , ๐œ ( ๐œŽ ) ) โ†’ ( ๐‘Œ , ๐œ ( ๐œ ) ) is ( ๐ผ , ๐ผ ) -fuzzy measurable.

Proof. Necessity. Suppose that ๐‘“ โˆถ ( ๐‘‹ , ๐œŽ ) โ†’ ( ๐‘Œ , ๐œ ) is ๐ผ -fuzzifying measurable. Then ๐œŽ ( ๐‘“ โˆ’ 1 ( ๐ด ) ) โ‰ฅ ๐œ ( ๐ด ) for any ๐ด โˆˆ ๐Ÿ ๐‘‹ . In order to prove that ๐‘“ โˆถ ( ๐‘‹ , ๐œ ( ๐œŽ ) ) โ†’ ( ๐‘Œ , ๐œ ( ๐œ ) ) is ( ๐ผ , ๐ผ ) -fuzzy measurable, we need to prove that ๐œ ( ๐œŽ ) ( ๐‘“ โ† ๐ฟ ( ๐ด ) ) โ‰ฅ ๐œ ( ๐œ ) ( ๐ด ) for any ๐ด โˆˆ ๐ผ ๐‘‹ .
In fact, for any ๐ด โˆˆ ๐ผ ๐‘‹ , by ๎€ท ๐‘“ ๐œ ( ๐œŽ ) โ† ๐ฟ ๎€ธ = ๎— ( ๐ด ) ๐ต โˆˆ โ„ฌ ๐œŽ ๎‚€ ๎€ท ๐‘“ โ† ๐ฟ ๎€ธ ( ๐ด ) โˆ’ 1 ๎‚ = ๎— ( ๐ต ) ๐ต โˆˆ โ„ฌ ๐œŽ ๎€ท ( ๐ด โˆ˜ ๐‘“ ) โˆ’ 1 ๎€ธ = ๎— ( ๐ต ) ๐ต โˆˆ โ„ฌ ๎— ๐œŽ ( ๐ต โˆ˜ ๐ด โˆ˜ ๐‘“ ) = ๐ต โˆˆ โ„ฌ ๐œŽ ๎€ท ๐‘“ โˆ’ 1 ๎€ท ๐ด โˆ’ 1 โ‰ฅ ๎— ( ๐ต ) ๎€ธ ๎€ธ ๐ต โˆˆ โ„ฌ ๐œ ๎€ท ๐ด โˆ’ 1 ๎€ธ ( ๐ต ) = ๐œ ( ๐œ ) ( ๐ด ) , ( 4 . 5 ) we can prove the necessity.
Sufficiency. Suppose that ๐‘“ โˆถ ( ๐‘‹ , ๐œ ( ๐œŽ ) ) โ†’ ( ๐‘Œ , ๐œ ( ๐œ ) ) is ( ๐ผ , ๐ผ ) -fuzzy measurable. Then ๐œ ( ๐œŽ ) ( ๐‘“ โ† ๐ผ ( ๐ด ) ) โ‰ฅ ๐œ ( ๐œ ) ( ๐ด ) for any ๐ด โˆˆ ๐ผ ๐‘‹ . In particular, it follows that ๐œ ( ๐œŽ ) ( ๐‘“ โ† ๐ผ ( ๐ด ) ) โ‰ฅ ๐œ ( ๐œ ) ( ๐ด ) for any ๐ด โˆˆ ๐Ÿ ๐‘‹ . In order to prove that ๐‘“ โˆถ ( ๐‘‹ , ๐œŽ ) โ†’ ( ๐‘Œ , ๐œ ) is ๐ผ -fuzzifying measurable, we need to prove that ๐œŽ ( ๐‘“ โˆ’ 1 ( ๐ด ) ) โ‰ฅ ๐œ ( ๐ด ) for any ๐ด โˆˆ ๐Ÿ ๐‘‹ . In fact, for any ๐ด โˆˆ ๐Ÿ ๐‘‹ and for any ๐ต โˆˆ โ„ฌ , if 0 , 1 โˆˆ ๐ต , then ๐ด โˆ’ 1 ( ๐ต ) = ๐‘‹ ; if 0 , 1 ยจ ยจ โˆˆ ๐ต , then ๐ด โˆ’ 1 ( ๐ต ) = โˆ… ; if only one of 0 and 1 is in ๐ต , then ๐ด โˆ’ 1 ( ๐ต ) = ๐ด or ๐ด โˆ’ 1 ( ๐ต ) = ๐ด โ€ฒ . However, we have ๐œŽ ๎€ท ๐‘“ โ† ๐ผ ๎€ธ ๎€ท ๐‘“ ( ๐ด ) = ๐œŽ โ† ๐ผ ๎€ธ ๎€ท ๐‘“ ( ๐ด ) = ๐œŽ โ† ๐ผ ( ๎€ธ ๎€ท ๐‘“ ๐ด ) โˆง ๐œŽ โ† ๐ผ ( ๐ด ) ๎…ž ๎€ธ = ๎— ๐ต โˆˆ โ„ฌ ๐œŽ ๎‚€ ๎€ท ๐‘“ โ† ๐ฟ ๎€ธ ( ๐ด ) โˆ’ 1 ( ๎‚ ๎€ท ๐‘“ ๐ต ) = ๐œ ( ๐œŽ ) โ† ๐ฟ ๎€ธ ๎€ท ๐ด ( ๐ด ) โ‰ฅ ๐œ ( ๐œ ) ( ๐ด ) = ๐œ ( ๐œ ) ( ๐ด ) โˆง ๐œ ( ๐œ ) ๎…ž ๎€ธ = ๎— ๐ต โˆˆ โ„ฌ ๐œ ๎€ท ๐ด โˆ’ 1 ๎€ธ ( ๐ต ) = ๐œ ( ๐ด ) . ( 4 . 6 )
This shows that ๐‘“ โˆถ ( ๐‘‹ , ๐œŽ ) โ†’ ( ๐‘Œ , ๐œ ) is ๐ผ -fuzzifying measurable.

Corollary 4.4. Let ( ๐‘‹ , ๐œŽ ) and ( ๐‘Œ , ๐œ ) be two measurable spaces, and ๐‘“ โˆถ ๐‘‹ โ†’ ๐‘Œ is a mapping. Then ๐‘“ โˆถ ( ๐‘‹ , ๐œŽ ) โ†’ ( ๐‘Œ , ๐œ ) is measurable if and only if ๐‘“ โˆถ ( ๐‘‹ , ๐œ ( ๐œŽ ) ) โ†’ ( ๐‘Œ , ๐œ ( ๐œ ) ) is ๐ผ -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).