International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 356581 | https://doi.org/10.1155/2010/356581

Fu-Gui Shi, " ( ๐ฟ , ๐‘€ ) -Fuzzy ๐œŽ -Algebras", International Journal of Mathematics and Mathematical Sciences, vol. 2010, Article ID 356581, 11 pages, 2010. https://doi.org/10.1155/2010/356581

( ๐ฟ , ๐‘€ ) -Fuzzy ๐œŽ -Algebras

Academic Editor: Andrei Volodin
Received23 Aug 2009
Revised19 Dec 2009
Accepted19 Jan 2010
Published24 Feb 2010

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).

References

  1. E. P. Klement, โ€œFuzzy ฯƒ-algebras and fuzzy measurable functions,โ€ Fuzzy Sets and Systems, vol. 4, no. 1, pp. 83โ€“93, 1980. View at: Publisher Site | Google Scholar | MathSciNet
  2. L. Biacino and A. Lettieri, โ€œL-ฯƒ-algebras and L-measures,โ€ Fuzzy Sets and Systems, vol. 44, no. 2, pp. 219โ€“225, 1991. View at: Publisher Site | Google Scholar | MathSciNet
  3. Ph. Dwinger, โ€œCharacterization of the complete homomorphic images of a completely distributive complete lattice. I,โ€ Indagationes Mathematicae, vol. 44, no. 4, pp. 403โ€“414, 1982. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  4. G.-J. Wang, โ€œTheory of topological molecular lattices,โ€ Fuzzy Sets and Systems, vol. 47, no. 3, pp. 351โ€“376, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. H.-L. Huang and F.-G. Shi, โ€œL-fuzzy numbers and their properties,โ€ Information Sciences, vol. 178, no. 4, pp. 1141โ€“1151, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. C. V. Negoiลฃฤƒ and D. A. Ralescu, Applications of Fuzzy Sets to Systems Analysis, vol. 11 of Interdisciplinary Systems Research Series, Birkhรคuser, Basel, Switzerland; Stuttgart and Halsted Press, New York, NY, USA, 1975. View at: MathSciNet
  7. F.-G. Shi, โ€œThe theory and applications of Lฮฒ-nested sets and Lฮฑ-nested sets and its applications,โ€ Fuzzy Systems and Mathematics, vol. 9, no. 4, pp. 65โ€“72, 1995 (Chinese). View at: Google Scholar | MathSciNet
  8. F.-G. Shi, โ€œL-fuzzy sets and prime element nested sets,โ€ Journal of Mathematical Research and Exposition, vol. 16, no. 3, pp. 398โ€“402, 1996. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  9. F.-G. Shi, โ€œTheory of molecular nested sets and its applications,โ€ Yantai Normal University Journal, vol. 1, pp. 33โ€“36, 1996 (Chinese). View at: Google Scholar
  10. F.-G. Shi, โ€œL-fuzzy relations and L-fuzzy subgroups,โ€ Journal of Fuzzy Mathematics, vol. 8, no. 2, pp. 491โ€“499, 2000. View at: Google Scholar | MathSciNet

Copyright © 2010 Fu-Gui Shi. 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.

Related articles

No related content is available yet for this article.
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views1722
Downloads729
Citations

Related articles

No related content is available yet for this article.

Article of the Year Award: Outstanding research contributions of 2021, as selected by our Chief Editors. Read the winning articles.