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 | 11 pages | https://doi.org/10.1155/2010/356581

( 𝐿 , 𝑀 ) -Fuzzy 𝜎 -Algebras

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

Abstract

The notion of ( 𝐿 , 𝑀 )-fuzzy 𝜎 -algebras is introduced in the lattice value fuzzy set theory. It is a generalization of Klement's fuzzy 𝜎 -algebras. In our definition of ( 𝐿 , 𝑀 )-fuzzy 𝜎 -algebras, each 𝐿 -fuzzy subset can be regarded as an 𝐿 -measurable set to some degree.

1. Introduction and Preliminaries

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

In this paper, when both 𝐿 and 𝑀 are complete lattices, we define an ( 𝐿 , 𝑀 ) -fuzzy 𝜎 -algebra on a nonempty set 𝑋 by means of a mapping 𝜎 ∢ 𝐿 𝑋 β†’ 𝑀 satisfying three axioms. Thus each 𝐿 -fuzzy subset of 𝑋 can be regarded as an 𝐿 -measurable set to some degree.

When 𝜎 is an ( 𝐿 , 𝑀 ) -fuzzy 𝜎 -algebra on 𝑋 , ( 𝑋 , 𝜎 ) is called an ( 𝐿 , 𝑀 ) -fuzzy measurable space. An ( 𝐿 , 2 ) -fuzzy 𝜎 -algebra is also called an 𝐿 - 𝜎 -algebra. A Klement 𝜎 -algebra can be viewed as a stratified [ 0 , 1 ] - 𝜎 -algebra. A Biacino-Lettieri 𝐿 - 𝜎 -algebra can be viewed as a stratified 𝐿 - 𝜎 -algebra. A ( 2 , 𝑀 ) -fuzzy 𝜎 -algebra is also called an 𝑀 -fuzzifying 𝜎 -algebra. A crisp 𝜎 -algebra can be regarded as a ( 2 , 2 ) -fuzzy 𝜎 -algebra.

Throughout this paper, both 𝐿 and 𝑀 denote complete lattices, and 𝐿 has an order-reversing involution β€² . 𝑋 is a nonempty set. 𝐿 𝑋 is the set of all 𝐿 -fuzzy sets (or 𝐿 -sets for short) on 𝑋 . We often do not distinguish a crisp subset 𝐴 of 𝑋 and its character function πœ’ 𝐴 . The smallest element and the largest element in 𝑀 are denoted by βŸ‚ 𝑀 and ⊀ 𝑀 , respectively.

The binary relation β‰Ί in 𝑀 is defined as follows: for π‘Ž , 𝑏 ∈ 𝑀 , π‘Ž β‰Ί 𝑏 if and only if for every subset 𝐷 βŠ† 𝑀 , the relation 𝑏 β©½ s u p 𝐷 always implies the existence of 𝑑 ∈ 𝐷 with π‘Ž β©½ 𝑑 [3]. { π‘Ž ∈ 𝑀 ∢ π‘Ž β‰Ί 𝑏 } is called the greatest minimal family of 𝑏 in the sense of [4], denoted by 𝛽 ( 𝑏 ) . Moreover, for 𝑏 ∈ 𝑀 , we define 𝛼 ( 𝑏 ) = { π‘Ž ∈ 𝑀 ∢ π‘Ž β‰Ί π‘œ 𝑝 𝑏 } . In a completely distributive lattice 𝑀 , there exist 𝛼 ( 𝑏 ) and 𝛽 ( 𝑏 ) for each 𝑏 ∈ 𝑀 , and ⋁ β‹€ 𝑏 = 𝛽 ( 𝑏 ) = 𝛼 ( 𝑏 ) (see [4]).

In [4], Wang thought that 𝛽 ( 0 ) = { 0 } and 𝛼 ( 1 ) = { 1 } . In fact, it should be that 𝛽 ( 0 ) = βˆ… and 𝛼 ( 1 ) = βˆ… .

For a complete lattice 𝐿 , 𝐴 ∈ 𝐿 𝑋 and π‘Ž ∈ 𝐿 , we use the following notation:

𝐴 [ π‘Ž ] = { π‘₯ ∈ 𝑋 ∢ 𝐴 ( π‘₯ ) β©Ύ π‘Ž } . ( 1 . 1 )

If 𝐿 is completely distributive, then we can define

𝐴 [ π‘Ž ] = { π‘₯ ∈ 𝑋 ∢ π‘Ž Β¨ Β¨ ∈ 𝛼 ( 𝐴 ( π‘₯ ) ) } . ( 1 . 2 )

Some properties of these cut sets can be found in [5–10].

Theorem 1.1 (see [4]). Let 𝑀 be a completely distributive lattice and { π‘Ž 𝑖 ∢ 𝑖 ∈ Ξ© } βŠ† 𝑀 . Then (1) β‹€ 𝛼 ( 𝑖 ∈ Ξ© π‘Ž 𝑖 ⋃ ) = 𝑖 ∈ Ξ© 𝛼 ( π‘Ž 𝑖 ) , that is, 𝛼 is an β‹€ βˆ’ ⋃ map;(2) ⋁ 𝛽 ( 𝑖 ∈ Ξ© π‘Ž 𝑖 ⋃ ) = 𝑖 ∈ Ξ© 𝛽 ( π‘Ž 𝑖 ) , that is, 𝛽 is a union-preserving map. For π‘Ž ∈ 𝐿 and 𝐷 βŠ† 𝑋 , we define two 𝐿 -fuzzy sets π‘Ž β‹€ 𝐷 and π‘Ž ∨ 𝐷 as follows: ξƒ― ( π‘Ž ∧ 𝐷 ) ( π‘₯ ) = π‘Ž , π‘₯ ∈ 𝐷 ; 0 , π‘₯ Β¨ Β¨ ξƒ― ∈ 𝐷 . ( π‘Ž ∨ 𝐷 ) ( π‘₯ ) = 1 , π‘₯ ∈ 𝐷 ; π‘Ž , π‘₯ Β¨ Β¨ ∈ 𝐷 . ( 1 . 3 ) Then for each 𝐿 -fuzzy set 𝐴 in 𝐿 𝑋 , it follows that  𝐴 = π‘Ž ∈ 𝐿 ξ€· π‘Ž ∧ 𝐴 [ π‘Ž ] ξ€Έ . ( 1 . 4 )

Theorem 1.2 (see [5, 7, 10]). If 𝐿 is completely distributive, then for each 𝐿 -fuzzy set 𝐴 in 𝐿 𝑋 , we have (1) ⋁ 𝐴 = π‘Ž ∈ 𝐿 β‹€ 𝐴 ( π‘Ž [ π‘Ž ] β‹€ ) = π‘Ž ∈ 𝐿 ( π‘Ž ∨ 𝐴 [ π‘Ž ] ) ; (2) f o r a l l π‘Ž ∈ 𝐿 , 𝐴 [ π‘Ž ] = β‹‚ 𝑏 ∈ 𝛽 ( π‘Ž ) 𝐴 [ 𝑏 ] ; (3) f o r a l l π‘Ž ∈ 𝐿 , 𝐴 [ π‘Ž ] = β‹‚ π‘Ž ∈ 𝛼 ( 𝑏 ) 𝐴 [ 𝑏 ] . For a family of 𝐿 -fuzzy sets { 𝐴 𝑖 ∢ 𝑖 ∈ Ξ© } in 𝐿 𝑋 , it is easy to see that   𝑖 ∈ Ξ© 𝐴 𝑖 ξƒͺ [ π‘Ž ] =  𝑖 ∈ Ξ© ξ€· 𝐴 𝑖 ξ€Έ [ π‘Ž ] . ( 1 . 5 ) If 𝐿 is completely distributive, then it follows [7] that   𝑖 ∈ Ξ© 𝐴 𝑖 ξƒͺ [ π‘Ž ] =  𝑖 ∈ Ξ© ξ€· 𝐴 𝑖 ξ€Έ [ π‘Ž ] . ( 1 . 6 )

Definition 1.3. Let 𝑋 be a nonempty set. A subset 𝜎 of [ 0 , 1 ] 𝑋 is called a Klement fuzzy 𝜎 -algebra if it satisfies the following three conditions:(1)for any constant fuzzy set 𝛼 , 𝛼 ∈ 𝜎 ;(2)for any 𝐴 ∈ [ 0 , 1 ] 𝑋 , 1 βˆ’ 𝐴 ∈ 𝜎 ;(3)for any { 𝐴 𝑛 ∢ 𝑛 ∈ β„• } βŠ† 𝜎 , ⋁ 𝑛 ∈ β„• 𝐴 𝑛 ∈ 𝜎 .The fuzzy sets in 𝜎 are called fuzzy measurable sets, and the pair ( 𝑋 , 𝜎 ) a fuzzy measurable space.

Definition 1.4. Let 𝐿 be a complete lattice with an order-reversing involution ξ…ž and 𝑋 a nonempty set. A subset 𝜎 of 𝐿 𝑋 is called an 𝐿 - 𝜎 -algebra if it satisfies the following three conditions:(1)for any π‘Ž ∈ 𝐿 , constant 𝐿 -fuzzy set π‘Ž β‹€ πœ’ 𝑋 ∈ 𝜎 ;(2)for any 𝐴 ∈ 𝐿 𝑋 , 𝐴 ξ…ž ∈ 𝜎 ;(3)for any { 𝐴 𝑛 ∢ 𝑛 ∈ β„• } βŠ† 𝜎 , ⋁ 𝑛 ∈ β„• 𝐴 𝑛 ∈ 𝜎 .The 𝐿 -fuzzy sets in 𝜎 are called 𝐿 -measurable sets, and the pair ( 𝑋 , 𝜎 ) an 𝐿 -measurable space.

2. ( 𝐿 , 𝑀 ) -Fuzzy 𝜎 -Algebras

L. Biacino and A. Lettieri defined that an 𝐿 - 𝜎 -algebra 𝜎 is a crisp subset of 𝐿 𝑋 . Now we consider an 𝑀 -fuzzy subset 𝜎 of 𝐿 𝑋 .

Definition. Let 𝑋 be a nonempty set. A mapping 𝜎 ∢ 𝐿 𝑋 β†’ 𝑀 is called an ( 𝐿 , 𝑀 ) -fuzzy 𝜎 -algebra if it satisfies the following three conditions: (LMS1) 𝜎 ( πœ’ βˆ… ) = ⊀ 𝑀 ;(LMS2) for any 𝐴 ∈ 𝐿 𝑋 , 𝜎 ( 𝐴 ) = 𝜎 ( 𝐴 ξ…ž ) ;(LMS3) for any { 𝐴 𝑛 ∢ 𝑛 ∈ β„• } βŠ† 𝐿 𝑋 , ⋁ 𝜎 ( 𝑛 ∈ β„• 𝐴 𝑛 β‹€ ) β‰₯ 𝑛 ∈ β„• 𝜎 ( 𝐴 𝑛 ) .An ( 𝐿 , 𝑀 ) -fuzzy 𝜎 -algebra 𝜎 is said to be stratified if and only if it satisfies the following condition:
(LMS1)*    βˆ€ π‘Ž ∈ 𝐿 , β‹€ πœ’ 𝜎 ( π‘Ž 𝑋 ) = ⊀ 𝑀 .
If 𝜎 is an ( 𝐿 , 𝑀 ) -fuzzy 𝜎 -algebra, then ( 𝑋 , 𝜎 ) is called an ( 𝐿 , 𝑀 ) -fuzzy measurable space.
An ( 𝐿 , 𝟐 ) -fuzzy 𝜎 -algebra is also called an 𝐿 - 𝜎 -algebra, and an ( 𝐿 , 𝟐 ) -fuzzy measurable space is also called an 𝐿 -measurable space.
A ( 𝟐 , 𝑀 ) -fuzzy 𝜎 -algebra is also called an 𝑀 -fuzzifying 𝜎 -algebra, and a ( 𝟐 , 𝑀 ) -fuzzy measurable space is also called an 𝑀 -fuzzifying measurable space.
Obviously a crisp measurable space can be regarded as a ( 𝟐 , 𝟐 ) -fuzzy measurable space.
If 𝜎 is an ( 𝐿 , 𝑀 ) -fuzzy 𝜎 -algebra, then 𝜎 ( 𝐴 ) can be regarded as the degree to which 𝐴 is an 𝐿 -measurable set.

Remark 2.2. If a subset 𝜎 of 𝐿 𝑋 is regarded as a mapping 𝜎 ∢ 𝐿 𝑋 β†’ 𝟐 , then 𝜎 is an 𝐿 - 𝜎 -algebra if and only if it satisfies the following conditions:(LS1) πœ’ βˆ… ∈ 𝜎 ;(LS2) 𝐴 ∈ 𝜎 β‡’ 𝐴 ξ…ž ∈ 𝜎 ;(LS3) for any { 𝐴 𝑛 ∢ 𝑛 ∈ β„• } βŠ† 𝜎 , ⋁ 𝑛 ∈ β„• 𝐴 𝑛 ∈ 𝜎 . Thus we easily see that a Klement 𝜎 -algebra is exactly a stratified [ 0 , 1 ] - 𝜎 -algebra, and a Biacino-Lettieri 𝐿 - 𝜎 -algebra is exactly a stratified 𝐿 - 𝜎 -algebra.
Moreover, when 𝐿 = 𝟐 , a mapping 𝜎 ∢ 𝟐 𝑋 β†’ 𝑀 is an 𝑀 -fuzzifying 𝜎 -algebra if and only if it satisfies the following conditions:
(MS1) 𝜎 ( βˆ… ) = ⊀ 𝑀 ;(MS2) for any 𝐴 ∈ 𝟐 𝑋 , 𝜎 ( 𝐴 ) = 𝜎 ( 𝐴 ξ…ž ) ;(MS3) for any { 𝐴 𝑛 ∢ 𝑛 ∈ β„• } βŠ† 2 𝑋 , ⋁ 𝜎 ( 𝑛 ∈ β„• 𝐴 𝑛 β‹€ ) β‰₯ 𝑛 ∈ β„• 𝜎 ( 𝐴 𝑛 ) .

Example 2.3. Let   ( 𝑋 , 𝜎 ) be a crisp measurable space. Define πœ’ 𝜎 ∢ 𝟐 𝑋 β†’ [ 0 , 1 ] by πœ’ 𝜎 ξƒ― ( 𝐴 ) = 1 , 𝐴 ∈ 𝜎 ; 0 , 𝐴 Β¨ Β¨ ∈ 𝜎 . ( 2 . 1 ) Then it is easy to prove that ( 𝑋 , πœ’ 𝜎 ) is a [ 0 , 1 ] -fuzzifying measurable space.

Example 2.4. Let 𝑋 be a nonempty set and 𝜎 ∢ 𝟐 𝑋 β†’ [ 0 , 1 ] a mapping defined by ξƒ― 𝜎 ( 𝐴 ) = 1 , 𝐴 ∈ { βˆ… , 𝑋 } ; 0 . 5 , 𝐴 Β¨ Β¨ ∈ { βˆ… , 𝑋 } . ( 2 . 2 ) Then it is easy to prove that ( 𝑋 , 𝜎 ) is a [ 0 , 1 ] -fuzzifying measurable space. If 𝐴 ∈ 𝟐 𝑋 with 𝐴 Β¨ Β¨ ∈ { βˆ… , 𝑋 } , then 0.5 is the degree to which 𝐴 is measurable.

Example 2.5. Let 𝑋 be a nonempty set and 𝜎 ∢ [ 0 , 1 ] 𝑋 β†’ [ 0 , 1 ] a mapping defined by ξƒ― ξ€½ πœ’ 𝜎 ( 𝐴 ) = 1 , 𝐴 ∈ βˆ… , πœ’ 𝑋 ξ€Ύ ; 0 . 5 , 𝐴 Β¨ Β¨ ∈ ξ€½ πœ’ βˆ… , πœ’ 𝑋 ξ€Ύ . ( 2 . 3 ) Then it is easy to prove that ( 𝑋 , 𝜎 ) is a ( [ 0 , 1 ] , [ 0 , 1 ] ) -fuzzy measurable space. If 𝐴 ∈ [ 0 , 1 ] 𝑋 with 𝐴 Β¨ Β¨ ∈ { πœ’ βˆ… , πœ’ 𝑋 } , then 0.5 is the degree to which 𝐴 is [ 0 , 1 ] -measurable.

Proposition 2.6. Let ( 𝑋 , 𝜎 ) be an ( 𝐿 , 𝑀 ) -fuzzy measurable spaces. Then for any { 𝐴 𝑛 ∢ 𝑛 ∈ β„• } βŠ† 𝐿 𝑋 , β‹€ 𝜎 ( 𝑛 ∈ β„• 𝐴 𝑛 β‹€ ) β‰₯ 𝑛 ∈ β„• 𝜎 ( 𝐴 𝑛 ) .

Proof. This can be proved from the following fact: 𝜎   𝑛 ∈ β„• 𝐴 𝑛 ξƒͺ   = 𝜎 𝑛 ∈ β„• ξ€· 𝐴 𝑛 ξ€Έ ξ…ž ξƒͺ β‰₯  𝑛 ∈ β„• 𝜎 ξ‚€ ξ€· 𝐴 𝑛 ξ€Έ ξ…ž  =  𝑛 ∈ β„• 𝜎 ξ€· 𝐴 𝑛 ξ€Έ . ( 2 . 4 ) The next two theorems give characterizations of an ( 𝐿 , 𝑀 ) -fuzzy 𝜎 -algebra.

Theorem 2.7. A mapping 𝜎 ∢ 𝐿 𝑋 β†’ 𝑀 is an ( 𝐿 , 𝑀 ) -fuzzy 𝜎 -algebra if and only if for each π‘Ž ∈ 𝑀 ⧡ { βŸ‚ 𝑀 } , 𝜎 [ π‘Ž ] is an 𝐿 - 𝜎 -algebra.

Proof. The proof is obvious and is omitted.

Corollary 2.8. A mapping 𝜎 ∢ 𝟐 𝑋 β†’ 𝑀 is an 𝑀 -fuzzifying 𝜎 -algebra if and only if for each π‘Ž ∈ 𝑀 ⧡ { βŸ‚ 𝑀 } , 𝜎 [ π‘Ž ] is a 𝜎 -algebra.

Theorem 2.9. If 𝑀 is completely distributive, then a mapping 𝜎 ∢ 𝐿 𝑋 β†’ 𝑀 is an ( 𝐿 , 𝑀 ) -fuzzy 𝜎 -algebra if and only if for each π‘Ž ∈ 𝛼 ( βŸ‚ 𝑀 ) , 𝜎 [ π‘Ž ] is an 𝐿 - 𝜎 -algebra.

Proof. Necessity.. Suppose that 𝜎 ∢ 𝐿 𝑋 β†’ 𝑀 is an ( 𝐿 , 𝑀 ) -fuzzy 𝜎 -algebra and π‘Ž ∈ 𝛼 ( βŸ‚ 𝑀 ) . Now we prove that 𝜎 [ π‘Ž ] is an 𝐿 - 𝜎 -algebra.(LS1) By 𝜎 ( πœ’ βˆ… ) = ⊀ 𝑀 and 𝛼 ( ⊀ 𝑀 ) = βˆ… , we know that π‘Ž Β¨ Β¨ ∈ 𝛼 ( 𝜎 ( πœ’ βˆ… ) ) ; this implies that πœ’ βˆ… ∈ 𝜎 [ π‘Ž ] .(LS2) If 𝐴 ∈ 𝜎 [ π‘Ž ] , then π‘Ž Β¨ Β¨ ∈ 𝛼 ( 𝜎 ( 𝐴 ) ) = 𝛼 ( 𝜎 ( 𝐴 ξ…ž ) ) ; this shows that 𝐴 ξ…ž ∈ 𝜎 [ π‘Ž ] .(LS3) If { 𝐴 𝑖 ∢ 𝑖 ∈ Ξ© } βŠ† 𝜎 [ π‘Ž ] , then f o r a l l 𝑖 ∈ Ξ© , π‘Ž Β¨ Β¨ ∈ 𝛼 ( 𝜎 ( 𝐴 𝑖 ) ) . Hence π‘Ž Β¨ Β¨ ∈ ⋃ 𝑖 ∈ Ξ© 𝛼 ( 𝜎 ( 𝐴 𝑖 ) ) . By ⋁ 𝜎 ( 𝑖 ∈ Ξ© 𝐴 𝑖 β‹€ ) β‰₯ 𝑖 ∈ Ξ© 𝜎 ( 𝐴 𝑖 ) , we know that 𝛼  𝜎   𝑖 ∈ Ξ© 𝐴 𝑖   ξƒͺ ξƒͺ βŠ† 𝛼 𝑖 ∈ Ξ© 𝜎 ξ€· 𝐴 𝑖 ξ€Έ ξƒͺ =  𝑖 ∈ Ξ© 𝛼 ξ€· 𝜎 ξ€· 𝐴 𝑖 . ξ€Έ ξ€Έ ( 2 . 5 ) This shows that π‘Ž Β¨ Β¨ ⋁ ∈ 𝛼 ( 𝜎 ( 𝑖 ∈ Ξ© 𝐴 𝑖 ) ) . Therefore, ⋁ 𝑖 ∈ Ξ© 𝐴 𝑖 ∈ 𝜎 [ π‘Ž ] . The proof is completed.

Corollary 2.10. If 𝑀 is completely distributive, then a mapping 𝜎 ∢ 𝟐 𝑋 β†’ 𝑀 is an 𝑀 -fuzzifying 𝜎 -algebra if and only if for each π‘Ž ∈ 𝛼 ( βŸ‚ 𝑀 ) , 𝜎 [ π‘Ž ] is a 𝜎 -algebra.

Now we consider the conditions that a family of 𝐿 - 𝜎 -algebras forms an ( 𝐿 , 𝑀 ) -fuzzy 𝜎 -algebra. By Theorem 1.2, we can obtain the following result.

Corollary 2.11. If 𝑀 is completely distributive, and 𝜎 is an ( 𝐿 , 𝑀 ) -fuzzy 𝜎 -algebra, then (1) 𝜎 [ 𝑏 ] βŠ† 𝜎 [ π‘Ž ] for any π‘Ž , 𝑏 ∈ 𝑀 ⧡ { βŸ‚ 𝑀 } with π‘Ž ∈ 𝛽 ( 𝑏 ) ;(2) 𝜎 [ 𝑏 ] βŠ† 𝜎 [ π‘Ž ] for any π‘Ž , 𝑏 ∈ 𝛼 ( βŸ‚ 𝑀 ) with 𝑏 ∈ 𝛼 ( π‘Ž ) .

Theorem 2.12. Let 𝑀 be completely distributive, and let { 𝜎 π‘Ž ∢ π‘Ž ∈ 𝛼 ( βŸ‚ 𝑀 ) } be a family of 𝐿 - 𝜎 -algebras. If 𝜎 π‘Ž = β‹‚ { 𝜎 𝑏 ∢ π‘Ž ∈ 𝛼 ( 𝑏 ) } for all π‘Ž ∈ 𝛼 ( βŸ‚ 𝑀 ) , then there exists an ( 𝐿 , 𝑀 ) -fuzzy 𝜎 -algebra 𝜎 such that 𝜎 [ π‘Ž ] = 𝜎 π‘Ž .

Proof. Suppose that 𝜎 π‘Ž = β‹‚ { 𝜎 𝑏 ∢ π‘Ž ∈ 𝛼 ( 𝑏 ) } for all π‘Ž ∈ 𝛼 ( βŸ‚ 𝑀 ) . Define 𝜎 ∢ 𝐿 𝑋 β†’ 𝑀 by  𝜎 ( 𝐴 ) = π‘Ž ∈ 𝑀 ( π‘Ž ∨ 𝜎 π‘Ž  ( 𝐴 ) ) = { π‘Ž ∈ 𝑀 ∢ 𝐴 Β¨ Β¨ ∈ 𝜎 π‘Ž } . ( 2 . 6 ) By Theorem 1.2, we can obtain that 𝜎 [ π‘Ž ] = 𝜎 π‘Ž .

Corollary 2.13. Let 𝑀 be completely distributive, and let { 𝜎 π‘Ž ∢ π‘Ž ∈ 𝛼 ( βŸ‚ 𝑀 ) } be a family of 𝜎 -algebras. If 𝜎 π‘Ž = β‹‚ { 𝜎 𝑏 ∢ π‘Ž ∈ 𝛼 ( 𝑏 ) } for all π‘Ž ∈ 𝛼 ( βŸ‚ 𝑀 ) , then there exists an 𝑀 -fuzzifying 𝜎 -algebra 𝜎 such that 𝜎 [ π‘Ž ] = 𝜎 π‘Ž .

Theorem 2.14. Let 𝑀 be completely distributive, and let { 𝜎 π‘Ž ∢ π‘Ž ∈ 𝑀 ⧡ { βŸ‚ 𝑀 } } be a family of 𝐿 - 𝜎 -algebra. If 𝜎 π‘Ž = β‹‚ { 𝜎 𝑏 ∢ 𝑏 ∈ 𝛽 ( π‘Ž ) } for all π‘Ž ∈ 𝑀 ⧡ { βŸ‚ 𝑀 } , then there exists an ( 𝐿 , 𝑀 ) -fuzzy 𝜎 -algebra 𝜎 such that 𝜎 [ π‘Ž ] = 𝜎 π‘Ž .

Proof. Suppose that 𝜎 π‘Ž = β‹‚ { 𝜎 𝑏 ∢ 𝑏 ∈ 𝛽 ( π‘Ž ) } for all π‘Ž ∈ 𝑀 ⧡ { βŸ‚ 𝑀 } . Define 𝜎 ∢ 𝐿 𝑋 β†’ 𝑀 by  𝜎 ( 𝐴 ) = π‘Ž ∈ 𝑀 ξ€· π‘Ž ∧ 𝜎 π‘Ž ξ€Έ =  ξ€½ ( 𝐴 ) π‘Ž ∈ 𝑀 ∢ 𝐴 ∈ 𝜎 π‘Ž ξ€Ύ . ( 2 . 7 ) By Theorem 1.2, we can obtain 𝜎 [ π‘Ž ] = 𝜎 π‘Ž .

Corollary 2.15. Let 𝑀 be completely distributive, and let { 𝜎 π‘Ž ∢ π‘Ž ∈ 𝑀 ⧡ { βŸ‚ 𝑀 } } be a family of 𝜎 -algebra. If 𝜎 π‘Ž = β‹‚ { 𝜎 𝑏 ∢ 𝑏 ∈ 𝛽 ( π‘Ž ) } for all π‘Ž ∈ 𝑀 ⧡ { βŸ‚ 𝑀 } , then there exists an 𝑀 -fuzzifying 𝜎 -algebra 𝜎 such that 𝜎 [ π‘Ž ] = 𝜎 π‘Ž .

Theorem 2.16. Let { 𝜎 𝑖 ∢ 𝑖 ∈ Ξ© } be a family of ( 𝐿 , 𝑀 ) -fuzzy 𝜎 -algebra on 𝑋 . Then β‹€ 𝑖 ∈ Ξ© 𝜎 𝑖 is an ( 𝐿 , 𝑀 ) -fuzzy 𝜎 -algebra on 𝑋 , where β‹€ 𝑖 ∈ Ξ© 𝜎 𝑖 ∢ 𝐿 𝑋 β†’ 𝑀 is defined by ( β‹€ 𝑖 ∈ Ξ© 𝜎 𝑖 β‹€ ) ( 𝐴 ) = 𝑖 ∈ Ξ© 𝜎 𝑖 ( 𝐴 ) .

Proof. This is straightforward.

3. ( 𝐿 , 𝑀 ) -Fuzzy Measurable Functions

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

Theorem 3.1. Let ( π‘Œ , 𝜏 ) be an ( 𝐿 , 𝑀 ) -fuzzy measurable space and 𝑓 ∢ 𝑋 β†’ π‘Œ a mapping. Define a mapping 𝑓 ← 𝐿 ( 𝜏 ) ∢ 𝐿 𝑋 β†’ 𝑀 by f o r a l l 𝐴 ∈ 𝐿 𝑋 , 𝑓 ← 𝐿  ξ€½ ( 𝜏 ) ( 𝐴 ) = 𝜏 ( 𝐡 ) ∢ 𝑓 ← 𝐿 ξ€Ύ , ( 𝐡 ) = 𝐴 w h e r e βˆ€ π‘₯ ∈ 𝑋 , 𝑓 ← 𝐿 ( 𝐡 ) ( π‘₯ ) = 𝐡 ( 𝑓 ( π‘₯ ) ) . ( 3 . 1 ) Then ( 𝑋 , 𝑓 ← 𝐿 ( 𝜏 ) ) is an ( 𝐿 , 𝑀 ) -fuzzy measurable space.

Proof. (LMS1) holds from the following equality: 𝑓 ← 𝐿 ξ€· πœ’ ( 𝜏 ) βˆ… ξ€Έ =  ξ€½ 𝜏 ( 𝐡 ) ∢ 𝑓 ← 𝐿 ( 𝐡 ) = πœ’ βˆ… ξ€Ύ ξ€· πœ’ = 𝜏 βˆ… ξ€Έ = ⊀ 𝑀 . ( 3 . 2 )
(LMS2) can be shown from the following fact: f o r a l l 𝐴 ∈ 𝐿 𝑋 , 𝑓 ← 𝐿  ξ€½ ( 𝜏 ) ( 𝐴 ) = 𝜏 ( 𝐡 ) ∢ 𝑓 ← 𝐿 ξ€Ύ =  ξ€½ 𝜏 ξ€· 𝐡 ( 𝐡 ) = 𝐴 ξ…ž ξ€Έ ∢ 𝑓 ← 𝐿 ξ€· 𝐡 ξ…ž ξ€Έ = 𝑓 ← 𝐿 ( 𝐡 ) ξ…ž = 𝐴 ξ…ž ξ€Ύ = 𝑓 ← 𝐿 ξ€· 𝐴 ( 𝜏 ) ξ…ž ξ€Έ . ( 3 . 3 )
(LMS3) for any { 𝐴 𝑛 ∢ 𝑛 ∈ β„• } βŠ† 𝐿 𝑋 , by 𝑓 ← 𝐿   ( 𝜏 ) 𝑛 ∈ β„• 𝐴 𝑛 ξƒͺ =  ξƒ― 𝜏 ( 𝐡 ) ∢ 𝑓 ← 𝐿  ( 𝐡 ) = 𝑛 ∈ β„• 𝐴 𝑛 ξƒ° β‰₯  ξƒ― 𝜏   𝑛 ∈ β„• 𝐡 𝑛 ξƒͺ ∢ 𝑓 ← 𝐿 ξ€· 𝐡 𝑛 ξ€Έ = 𝐴 𝑛 ξƒ° β‰₯  𝑛 ∈ β„• 𝑓 ← 𝐿 ( ξ€· 𝐴 𝜏 ) 𝑛 ξ€Έ ( 3 . 4 ) w e c a n p r o v e ( L M S 3 ) .

Definition 3.2. Let ( 𝑋 , 𝜎 ) and ( π‘Œ , 𝜏 ) be ( 𝐿 , 𝑀 ) -fuzzy measurable spaces. A mapping 𝑓 ∢ 𝑋 β†’ π‘Œ is called ( 𝐿 , 𝑀 ) -fuzzy measurable if 𝜎 ( 𝑓 ← 𝐿 ( 𝐡 ) ) β‰₯ 𝜏 ( 𝐡 ) for all 𝐡 ∈ 𝐿 π‘Œ .
An ( 𝐿 , 𝟐 ) -fuzzy measurable mapping is called an 𝐿 -measurable mapping, and a ( 𝟐 , 𝑀 ) -fuzzy measurable mapping is called an 𝑀 -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 ( 𝐿 , 𝑀 ) -fuzzy measurable mappings.

Theorem 3.3. Let ( 𝑋 , 𝜎 ) and ( π‘Œ , 𝜏 ) be two ( 𝐿 , 𝑀 ) -fuzzy measurable spaces. A mapping 𝑓 ∢ 𝑋 β†’ π‘Œ is ( 𝐿 , 𝑀 ) -fuzzy measurable if and only if 𝑓 ← 𝐿 ( 𝜏 ) ( 𝐴 ) ≀ 𝜎 ( 𝐴 ) for all 𝐴 ∈ 𝐿 𝑋 .

Proof. Necessity.. If 𝑓 ∢ 𝑋 β†’ π‘Œ is ( 𝐿 , 𝑀 ) -fuzzy measurable, then 𝜎 ( 𝑓 ← 𝐿 ( 𝐡 ) ) β‰₯ 𝜏 ( 𝐡 ) for all 𝐡 ∈ 𝐿 π‘Œ . Hence for all 𝐡 ∈ 𝐿 π‘Œ , we have 𝑓 ← 𝐿  ξ€½ ( 𝜏 ) ( 𝐴 ) = 𝜏 ( 𝐡 ) ∢ 𝑓 ← 𝐿 ξ€Ύ ≀  ξ€½ 𝜎 ξ€· 𝑓 ( 𝐡 ) = 𝐴 ← 𝐿 ξ€Έ ( 𝐡 ) ∢ 𝑓 ← 𝐿 ξ€Ύ ( 𝐡 ) = 𝐴 = 𝜎 ( 𝐴 ) . ( 3 . 5 ) Sufficiency.. If 𝑓 ← 𝐿 ( 𝜏 ) ( 𝐴 ) ≀ 𝜎 ( 𝐴 ) for all 𝐴 ∈ 𝐿 𝑋 , then 𝜏 ( 𝐡 ) ≀ 𝑓 ← 𝐿 ( 𝜏 ) ( 𝑓 ← 𝐿 ( 𝐡 ) ) ≀ 𝜎 ( 𝑓 ← 𝐿 ( 𝐡 ) ) for all 𝐡 ∈ 𝐿 π‘Œ ; this shows that 𝑓 ∢ 𝑋 β†’ π‘Œ is ( 𝐿 , 𝑀 ) -fuzzy measurable.

The next three theorems are trivial.

Theorem 3.4. If 𝑓 ∢ ( 𝑋 , 𝜎 ) β†’ ( π‘Œ , 𝜏 ) and 𝑓 ∢ ( π‘Œ , 𝜏 ) β†’ ( 𝑍 , 𝜌 ) are ( 𝐿 , 𝑀 ) -fuzzy measurable, then 𝑔 ∘ 𝑓 ∢ ( 𝑋 , 𝜎 ) β†’ ( 𝑍 , 𝜌 ) is ( 𝐿 , 𝑀 ) -fuzzy measurable.

Theorem 3.5. Let ( 𝑋 , 𝜎 ) and ( π‘Œ , 𝜏 ) be ( 𝐿 , 𝑀 ) -fuzzy measurable spaces. Then a mapping 𝑓 ∢ ( 𝑋 , 𝜎 ) β†’ ( π‘Œ , 𝜏 ) is ( 𝐿 , 𝑀 ) -fuzzy measurable if and only if 𝑓 ∢ ( 𝑋 , 𝜎 [ π‘Ž ] ) β†’ ( π‘Œ , 𝜏 [ π‘Ž ] ) is 𝐿 -measurable for any π‘Ž ∈ 𝑀 ⧡ { βŸ‚ 𝑀 } .

Theorem 3.6. Let 𝑀 be completely distributive, and let ( 𝑋 , 𝜎 ) and ( π‘Œ , 𝜏 ) be ( 𝐿 , 𝑀 ) -fuzzy measurable spaces. Then a mapping 𝑓 ∢ ( 𝑋 , 𝜎 ) β†’ ( π‘Œ , 𝜏 ) is ( 𝐿 , 𝑀 ) -fuzzy measurable if and only if 𝑓 ∢ ( 𝑋 , 𝜎 [ π‘Ž ] ) β†’ ( π‘Œ , 𝜏 [ π‘Ž ] ) is 𝐿 -measurable for any π‘Ž ∈ 𝛼 ( βŸ‚ 𝑀 ) .

Corollary 3.7. Let ( 𝑋 , 𝜎 ) and ( π‘Œ , 𝜏 ) be 𝑀 -fuzzifying measurable spaces. Then a mapping 𝑓 ∢ ( 𝑋 , 𝜎 ) β†’ ( π‘Œ , 𝜏 ) is 𝑀 -fuzzifying measurable if and only if 𝑓 ∢ ( 𝑋 , 𝜎 [ π‘Ž ] ) β†’ ( π‘Œ , 𝜏 [ π‘Ž ] ) is measurable for any π‘Ž ∈ 𝑀 ⧡ { βŸ‚ 𝑀 } .

Corollary 3.8. Let 𝑀 be completely distributive, and let ( 𝑋 , 𝜎 ) and ( π‘Œ , 𝜏 ) be 𝑀 -fuzzifying measurable spaces. Then a mapping 𝑓 ∢ ( 𝑋 , 𝜎 ) β†’ ( π‘Œ , 𝜏 ) is 𝑀 -fuzzifying measurable if and only if 𝑓 ∢ ( 𝑋 , 𝜎 [ π‘Ž ] ) β†’ ( π‘Œ , 𝜏 [ π‘Ž ] ) is measurable for any π‘Ž ∈ 𝛼 ( βŸ‚ 𝑀 ) .

4. ( 𝐼 , 𝐼 ) -Fuzzy 𝜎 -Algebras Generated by 𝐼 -Fuzzifying 𝜎 -Algebras

In this section, ℬ will be used to denote the 𝜎 -algebra of Borel subsets of 𝐼 = [ 0 , 1 ] .

Theorem 4.1. Let ( 𝑋 , 𝜎 ) be an 𝐼 -fuzzifying measurable space. Define a mapping 𝜁 ( 𝜎 ) ∢ 𝐼 𝑋 β†’ 𝐼 by  𝜁 ( 𝜎 ) ( 𝐴 ) = 𝐡 ∈ ℬ 𝜎 ξ€· 𝐴 βˆ’ 1 ξ€Έ . ( 𝐡 ) ( 4 . 1 ) Then 𝜁 ( 𝜎 ) is a stratified ( 𝐼 , 𝐼 ) -fuzzy 𝜎 -algebra, which is said to be the ( 𝐼 , 𝐼 ) -fuzzy 𝜎 -algebra generated by 𝜎 .

Proof. (LMS1) For any 𝐡 ∈ ℬ and for any π‘Ž ∈ 𝐼 , if π‘Ž ∈ 𝐡 , then β‹€ πœ’ ( π‘Ž 𝑋 ) βˆ’ 1 ( 𝐡 ) = 𝑋 ; if π‘Ž Β¨ Β¨ ∈ 𝐡 , then β‹€ πœ’ ( π‘Ž 𝑋 ) βˆ’ 1 ( 𝐡 ) = βˆ… . However, we have that β‹€ πœ’ 𝜎 ( ( π‘Ž 𝑋 ) βˆ’ 1 ( 𝐡 ) ) = 1 . This shows that β‹€ πœ’ 𝜁 ( 𝜎 ) ( π‘Ž 𝑋 ) = 1 .
(LMS2) f o r a l l 𝐴 ∈ 𝐼 𝑋 and f o r a l l 𝐡 ∈ ℬ , we have ξ€· 𝐴 𝜁 ( 𝜎 ) ξ…ž ξ€Έ =  𝐡 ∈ ℬ 𝜎 ξ€· ( 1 βˆ’ 𝐴 ) βˆ’ 1 ξ€Έ =  ( 𝐡 ) 𝐡 ∈ ℬ 𝜎 =  ( { π‘₯ ∈ 𝑋 ∢ 1 βˆ’ 𝐴 ( π‘₯ ) ∈ 𝐡 } ) 𝐡 ∈ ℬ 𝜎 ( { π‘₯ ∈ 𝑋 ∢ βˆƒ 𝑏 ∈ 𝐡 , s . t =  . 𝐴 ( π‘₯ ) = 1 βˆ’ 𝑏 } ) 𝐡 ∈ ℬ 𝜎 ξ€· 𝐴 βˆ’ 1 ξ€Έ ( 𝐡 ) = 𝜁 ( 𝜎 ) ( 𝐴 ) . ( 4 . 2 )
(LMS3) for any { 𝐴 𝑛 ∢ 𝑛 ∈ β„• } βŠ† 𝐿 𝑋 and f o r a l l 𝐡 ∈ ℬ , by 𝜁   ( 𝜎 ) 𝑛 ∈ β„• 𝐴 𝑛 ξƒͺ =  𝐡 ∈ ℬ 𝜎 βŽ› ⎜ ⎜ ⎝   𝑛 ∈ β„• 𝐴 𝑛 ξƒͺ βˆ’ 1 ⎞ ⎟ ⎟ ⎠ =  ( 𝐡 ) 𝐡 ∈ ℬ 𝜎   𝑛 ∈ β„• 𝐴 𝑛 βˆ’ 1 ξƒͺ β‰₯  ( 𝐡 ) 𝐡 ∈ ℬ  𝑛 ∈ β„• 𝜎 ξ€· 𝐴 𝑛 βˆ’ 1 ξ€Έ =  ( 𝐡 ) 𝑛 ∈ β„•  𝐡 ∈ ℬ 𝜎 ξ€· 𝐴 𝑛 βˆ’ 1 ξ€Έ =  ( 𝐡 ) 𝑛 ∈ β„• ξ€· 𝐴 𝜁 ( 𝜎 ) 𝑛 ξ€Έ , ( 4 . 3 )
we obtain ⋁ 𝜁 ( 𝜎 ) ( 𝑛 ∈ β„• 𝐴 𝑛 β‹€ ) β‰₯ 𝑛 ∈ β„• 𝜁 ( 𝜎 ) ( 𝐴 𝑛 ) .

Corollary 4.2. Let ( 𝑋 , 𝜎 ) be a measurable space. Define a subset 𝜁 ( 𝜎 ) βŠ† 𝐼 𝑋 ( can be viewed as a mapping 𝜁 ( 𝜎 ) ∢ 𝐼 𝑋 β†’ 𝟐 ) by ξ€½ 𝜁 ( 𝜎 ) = 𝐴 ∈ 𝐼 𝑋 ∢ βˆ€ 𝐡 ∈ ℬ , 𝐴 βˆ’ 1 ξ€Ύ . ( 𝐡 ) ∈ 𝜎 ( 4 . 4 ) Then 𝜁 ( 𝜎 ) is a stratified 𝐼 - 𝜎 -algebra.

From Corollary 4.2, we see that the functor 𝜁 in Theorem 4.1 is a generalization of Klement functor 𝜁 .

Theorem 4.3. Let ( 𝑋 , 𝜎 ) and ( π‘Œ , 𝜏 ) be two 𝐼 -fuzzifying measurable spaces, and 𝑓 ∢ 𝑋 β†’ π‘Œ is a map. Then 𝑓 ∢ ( 𝑋 , 𝜎 ) β†’ ( π‘Œ , 𝜏 ) is 𝐼 -fuzzifying measurable if and only if 𝑓 ∢ ( 𝑋 , 𝜁 ( 𝜎 ) ) β†’ ( π‘Œ , 𝜁 ( 𝜏 ) ) is ( 𝐼 , 𝐼 ) -fuzzy measurable.

Proof. Necessity. Suppose that 𝑓 ∢ ( 𝑋 , 𝜎 ) β†’ ( π‘Œ , 𝜏 ) is 𝐼 -fuzzifying measurable. Then 𝜎 ( 𝑓 βˆ’ 1 ( 𝐴 ) ) β‰₯ 𝜏 ( 𝐴 ) for any 𝐴 ∈ 𝟐 𝑋 . In order to prove that 𝑓 ∢ ( 𝑋 , 𝜁 ( 𝜎 ) ) β†’ ( π‘Œ , 𝜁 ( 𝜏 ) ) is ( 𝐼 , 𝐼 ) -fuzzy measurable, we need to prove that 𝜁 ( 𝜎 ) ( 𝑓 ← 𝐿 ( 𝐴 ) ) β‰₯ 𝜁 ( 𝜏 ) ( 𝐴 ) for any 𝐴 ∈ 𝐼 𝑋 .
In fact, for any 𝐴 ∈ 𝐼 𝑋 , by ξ€· 𝑓 𝜁 ( 𝜎 ) ← 𝐿 ξ€Έ =  ( 𝐴 ) 𝐡 ∈ ℬ 𝜎 ξ‚€ ξ€· 𝑓 ← 𝐿 ξ€Έ ( 𝐴 ) βˆ’ 1  =  ( 𝐡 ) 𝐡 ∈ ℬ 𝜎 ξ€· ( 𝐴 ∘ 𝑓 ) βˆ’ 1 ξ€Έ =  ( 𝐡 ) 𝐡 ∈ ℬ  𝜎 ( 𝐡 ∘ 𝐴 ∘ 𝑓 ) = 𝐡 ∈ ℬ 𝜎 ξ€· 𝑓 βˆ’ 1 ξ€· 𝐴 βˆ’ 1 β‰₯  ( 𝐡 ) ξ€Έ ξ€Έ 𝐡 ∈ ℬ 𝜏 ξ€· 𝐴 βˆ’ 1 ξ€Έ ( 𝐡 ) = 𝜁 ( 𝜏 ) ( 𝐴 ) , ( 4 . 5 ) we can prove the necessity.
Sufficiency. Suppose that 𝑓 ∢ ( 𝑋 , 𝜁 ( 𝜎 ) ) β†’ ( π‘Œ , 𝜁 ( 𝜏 ) ) is ( 𝐼 , 𝐼 ) -fuzzy measurable. Then 𝜁 ( 𝜎 ) ( 𝑓 ← 𝐼 ( 𝐴 ) ) β‰₯ 𝜁 ( 𝜏 ) ( 𝐴 ) for any 𝐴 ∈ 𝐼 𝑋 . In particular, it follows that 𝜁 ( 𝜎 ) ( 𝑓 ← 𝐼 ( 𝐴 ) ) β‰₯ 𝜁 ( 𝜏 ) ( 𝐴 ) for any 𝐴 ∈ 𝟐 𝑋 . In order to prove that 𝑓 ∢ ( 𝑋 , 𝜎 ) β†’ ( π‘Œ , 𝜏 ) is 𝐼 -fuzzifying measurable, we need to prove that 𝜎 ( 𝑓 βˆ’ 1 ( 𝐴 ) ) β‰₯ 𝜏 ( 𝐴 ) for any 𝐴 ∈ 𝟐 𝑋 . In fact, for any 𝐴 ∈ 𝟐 𝑋 and for any 𝐡 ∈ ℬ , if 0 , 1 ∈ 𝐡 , then 𝐴 βˆ’ 1 ( 𝐡 ) = 𝑋 ; if 0 , 1 Β¨ Β¨ ∈ 𝐡 , then 𝐴 βˆ’ 1 ( 𝐡 ) = βˆ… ; if only one of 0 and 1 is in 𝐡 , then 𝐴 βˆ’ 1 ( 𝐡 ) = 𝐴 or 𝐴 βˆ’ 1 ( 𝐡 ) = 𝐴 β€² . However, we have 𝜎 ξ€· 𝑓 ← 𝐼 ξ€Έ ξ€· 𝑓 ( 𝐴 ) = 𝜎 ← 𝐼 ξ€Έ ξ€· 𝑓 ( 𝐴 ) = 𝜎 ← 𝐼 ( ξ€Έ ξ€· 𝑓 𝐴 ) ∧ 𝜎 ← 𝐼 ( 𝐴 ) ξ…ž ξ€Έ =  𝐡 ∈ ℬ 𝜎 ξ‚€ ξ€· 𝑓 ← 𝐿 ξ€Έ ( 𝐴 ) βˆ’ 1 (  ξ€· 𝑓 𝐡 ) = 𝜁 ( 𝜎 ) ← 𝐿 ξ€Έ ξ€· 𝐴 ( 𝐴 ) β‰₯ 𝜁 ( 𝜏 ) ( 𝐴 ) = 𝜁 ( 𝜏 ) ( 𝐴 ) ∧ 𝜁 ( 𝜏 ) ξ…ž ξ€Έ =  𝐡 ∈ ℬ 𝜏 ξ€· 𝐴 βˆ’ 1 ξ€Έ ( 𝐡 ) = 𝜏 ( 𝐴 ) . ( 4 . 6 )
This shows that 𝑓 ∢ ( 𝑋 , 𝜎 ) β†’ ( π‘Œ , 𝜏 ) is 𝐼 -fuzzifying measurable.

Corollary 4.4. Let ( 𝑋 , 𝜎 ) and ( π‘Œ , 𝜏 ) be two measurable spaces, and 𝑓 ∢ 𝑋 β†’ π‘Œ is a mapping. Then 𝑓 ∢ ( 𝑋 , 𝜎 ) β†’ ( π‘Œ , 𝜏 ) is measurable if and only if 𝑓 ∢ ( 𝑋 , 𝜁 ( 𝜎 ) ) β†’ ( π‘Œ , 𝜁 ( 𝜏 ) ) is 𝐼 -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

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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.


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