Volume 2010, Article ID 356581,11pages doi:10.1155/2010/356581

*Research Article*

*L, M*

**-Fuzzy**

*σ*

**-Algebras**

**Fu-Gui Shi**

*Department of Mathematics, School of Science, Beijing Institute of Technology, Beijing 100081, China*

Correspondence should be addressed to Fu-Gui Shi,fuguishi@bit.edu.cn

Received 23 August 2009; Revised 19 December 2009; Accepted 19 January 2010

Academic Editor: Andrei Volodin

Copyrightq2010 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.

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 ofL, 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 in1in order to prepare
a measure theory for fuzzy sets. In the definition of Klement’s fuzzy*σ*-algebraX, 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 setting2.

In this paper, when both*L* and*M*are complete lattices, we define anL, M-fuzzy

*σ*-algebra on a nonempty set*X*by means of a mapping*σ*:*LX* _{→} _{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 anL, M-fuzzy*σ*-algebra on*X*,X, σis called anL, M-fuzzy measurable
space. AnL,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. A2*, M*-fuzzy*σ*-algebra is also called an*M*-fuzzifying*σ*-algebra. A
crisp*σ*-algebra can be regarded as a2*,*2-fuzzy*σ*-algebra.

Throughout this paper, both*L*and*M*denote complete lattices, and*L* has an
order-reversing involution’. *X* is a nonempty set. *LX* _{is the set of all} _{L}_{-fuzzy sets}_{}_{or} _{L}_{-sets for}

shorton*X*. We often do not distinguish a crisp subset*A*of*X*and its character function*χA*.
The smallest element and the largest element in*M*are denoted by⊥*M*and*M*, respectively.

*βb*. Moreover, for*b*∈ *M*, we define*αb * {*a*∈*M* : *a*≺*op _{b}*

_{}}

_{. In a completely distributive}

lattice*M*, there exist*αb*and*βb*for each*b*∈*M*, and*bβb αb *see4.

In4, Wang thought that*β*0 {0}and*α*1 {1}. In fact, it should be that*β*0 ∅
and*α*1 ∅.

For a complete lattice*L*,*A*∈*LX*and*a*∈*L*, we use the following notation:

*A*_{}*a*{*x*∈*X* :*Axa*}*.* 1.1

If*L*is completely distributive, then we can define

*Aa*_{}_{{}_{x}_{∈}_{X}_{:}_{a}

∈*αAx*}*.* 1.2

Some properties of these cut sets can be found in5–10.

**Theorem 1.1**see4*. LetMbe a completely distributive lattice and*{*ai*:*i*∈Ω} ⊆*M. Then*

1*α _{i}*

_{∈}

_{Ω}

*ai*

_{i}_{∈}

_{Ω}

*αai, that is,αis an*−

*map;*

2*β _{i}*

_{∈}

_{Ω}

*ai*

_{i}_{∈}

_{Ω}

*βai, that is,βis a union-preserving map.*

*Fora*∈

*LandD*⊆

*X, we define twoL-fuzzy setsa*∧

*Danda*∨

*Das follows:*

a∧*Dx *

⎧ ⎨ ⎩

*a, x*∈*D*;

0*,* *x*∈_{}*D.* a∨*Dx *

⎧ ⎨ ⎩

1*,* *x*∈*D*;

*a, x*∈_{}*D.* 1.3

*Then for eachL-fuzzy setAinLX _{, it follows that}*

*A*

*a*∈*L*

*a*∧*A*_{}*a.* _{}_{1.4}_{}

**Theorem 1.2**see5,7,10*. IfLis completely distributive, then for eachL-fuzzy setAinLX _{, we}*

*have*

1*A _{a}*

_{∈}

*a∧*

_{L}*Aa*

*a*∈*L*a∨*Aa*;
2*for alla*∈*L,Aa*

*b*∈*βaAb*;
3*for alla*∈*L,Aa*_{}

*a*∈*αbAb.*

*For a family ofL-fuzzy sets*{*Ai*:*i*∈Ω}*inLX, it is easy to see that*

*i*∈Ω

*Ai*

*a*

*i*∈Ω

Ai_{}*a.* 1.5

*IfLis completely distributive, then it follows [7] that*

*i*∈Ω

*Ai*

_{}*a*

*i*∈Ω

*Definition 1.3. Let* *X* be a nonempty set. A subset*σ* of0*,*1*X* is called a Klement fuzzy*σ*
-algebra if it satisfies the following three conditions:

1for any constant fuzzy set*α*,*α*∈*σ*;

2for any*A*∈0*,*1*X*, 1−*A*∈*σ*;

3for any{*An*:*n*∈N} ⊆*σ*,

*n*∈N*An*∈*σ*.

The fuzzy sets in*σ*are called fuzzy measurable sets, and the pairX, σa fuzzy measurable
space.

*Definition 1.4. Let* *L* be a complete lattice with an order-reversing involution and *X* a
nonempty set. A subset *σ* of *LX* _{is called an} _{L}_{-}_{σ}_{-algebra if it satisfies the following three}

conditions:

1for any*a*∈*L*, constant*L*-fuzzy set*a*∧*χX*∈*σ*;
2for any*A*∈*LX*_{,}_{A}_{∈}_{σ}_{;}

3for any{*An*:*n*∈N} ⊆*σ*,*n*∈N*An*∈*σ*.

The*L*-fuzzy sets in*σ*are called*L*-measurable sets, and the pairX, σan*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 *LX*_{. Now we}

consider an*M*-fuzzy subset*σ*of*LX*.

*Definition 2.1. LetX* be a nonempty set. A mapping*σ* : *LX* _{→} _{M}_{is called an}_{L, M}_{-fuzzy}
*σ*-algebra if it satisfies the following three conditions:

**LMS1***σχ*∅ *M*;

**LMS2**for any*A*∈*LX*_{,}_{σA }_{σA}_{}_{;}

**LMS3**for any{*An*:*n*∈N} ⊆*LX*_{,}_{σ}

*n*∈N *An*≥*n*∈N *σAn*.

An L, M-fuzzy *σ*-algebra*σ* is said to be stratified if and only if it satisfies the following
condition:

**LMS1**∗ _{∀}_{a}_{∈}_{L}_{,}_{σa}_{∧}_{χX }_{}
*M*.

If*σ* is an L, M-fuzzy *σ*-algebra, then X, σis called an L, M-fuzzy measurable
space.

AnL,**2**-fuzzy*σ*-algebra is also called an*L*-*σ*-algebra, and anL,**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 anL, M-fuzzy*σ*-algebra, then*σA*can be regarded as the degree to which*A*

*Remark 2.2. If a subsetσ*of*LX*_{is regarded as a mapping}_{σ}_{:}_{L}X_{→} _{2, then}_{σ}_{is an}_{L}_{-}_{σ}_{-algebra}

if and only if it satisfies the following conditions:

**LS1***χ*∅ ∈*σ*;

**LS2***A*∈*σ*⇒*A*∈*σ*;

**LS3**for any{*An*:*n*∈N} ⊆*σ*,_{n}_{∈}_{N}*An*∈*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* _{→} _{M}_{is an}_{M}_{-fuzzifying}_{σ}_{-algebra if and}

only if it satisfies the following conditions:

**MS1***σ*∅ *M*;

**MS2**for any*A*∈**2***X*_{,}_{σA }_{σA}_{}_{;}

**MS3**for any{*An*:*n*∈N} ⊆2*X*,*σ*

*n*∈N*An*≥

*n*∈N *σAn*.

*Example 2.3. Let*X, σbe a crisp measurable space. Define*χσ* :**2***X* → 0*,*1by

*χσA *

⎧ ⎨ ⎩

1*, A*∈*σ*;

0*, A*∈_{}*σ.*

2.1

Then it is easy to prove thatX, χ*σ*is a0*,*1-fuzzifying measurable space.

*Example 2.4. LetX*be a nonempty set and*σ*:**2***X* _{→} _{}_{0}_{,}_{1}_{}_{a mapping defined by}

*σA *

⎧ ⎨ ⎩

1*,* *A*∈ {∅*, X*};

0*.*5*, A*∈{∅_{} *, X*}*.* 2.2

Then it is easy to prove that X, σis a0*,*1-fuzzifying measurable space. If*A* ∈ **2***X* _{with}
*A*∈{∅_{} *, X*}, then 0.5 is the degree to which*A*is measurable.

*Example 2.5. LetX*be a nonempty set and*σ*:0*,*1*X* → 0*,*1a mapping defined by

*σA *

⎧ ⎨ ⎩

1*,* *A*∈*χ*∅*, χX*;

0*.*5*, A*∈_{}*χ*∅*, χX.*

Then it is easy to prove thatX, σis a0*,*1,0*,*1-fuzzy measurable space. If*A* ∈0*,*1*X*
with*A*∈{_{} *χ*_{∅}*, χX*}, then 0.5 is the degree to which*A*is0*,*1-measurable.

* Proposition 2.6. Let*X, σ

*be an*L, M

*-fuzzy measurable spaces. Then for any*{

*An*:

*n*∈N} ⊆

*LX,*

*σn*∈N

*An*≥

*n*∈N

*σAn.*

*Proof. This can be proved from the following fact:*

*σ*

*n*∈N
*An*

*σ*

*n*∈N
A*n*

≥

*n*∈N

*σ* A*n*
*n*∈N

*σAn.* 2.4

The next two theorems give characterizations of anL, M-fuzzy*σ*-algebra.

* Theorem 2.7. A mappingσ* :

*LX*

_{→}

_{M}_{is an}_{L, M}

_{-fuzzy}_{σ}_{-algebra if and only if for each}_{a}_{∈}

*M*\ {⊥

*M*}

*,σais anL-σ-algebra.*

*Proof. The proof is obvious and is omitted.*

* Corollary 2.8. A mappingσ* :

**2**

*X*

_{→}

_{M}_{is an}_{M}_{-fuzzifying}_{σ}_{-algebra if and only if for each}

_{a}_{∈}

*M*\ {⊥

*M*}

*,σais aσ-algebra.*

* Theorem 2.9. IfMis completely distributive, then a mappingσ* :

*LX*

_{→}

_{M}_{is an}_{L, M}

_{-fuzzy}*σ-algebra if and only if for eacha*∈

*α*⊥

*M,σa*

_{is an}_{L}_{-}_{σ}_{-algebra.}*Proof.*

*Necessity. Suppose thatσ* :*LX* _{→} _{M}_{is an}_{L, M}_{-fuzzy}_{σ}_{-algebra and}_{a}_{∈}_{α}_{⊥}_{M}_{. Now we}

prove that*σa*_{is an}_{L}_{-}_{σ}_{-algebra.}

**LS1**By*σχ*_{∅} *M*and*αM *∅, we know that*a*∈*ασχ*∅; this implies that*χ*∅∈*σa*.

**LS2**If*A*∈*σa*_{, then}_{a}_{∈}_{}_{ασA }_{ασA}_{}_{; this shows that}_{A}_{∈}_{σ}a_{.}

**LS3**If{*Ai* : *i* ∈Ω} ⊆ *σa*, then for all*i* ∈ Ω,*a*∈*ασAi*. Hence*a*∈

*i*∈Ω*ασAi*. By

*σi*∈Ω *Ai*≥*i*∈Ω*σAi*, we know that

*α*

*σ*

_{}

*i*∈Ω

*Ai*

⊆*α*

_{}

*i*∈Ω

*σAi*

*i*∈Ω

*ασAi.* 2.5

This shows that *a*∈_{}*ασ _{i}*

_{∈}

_{Ω}

*Ai*. Therefore,

_{i}_{∈}

_{Ω}

*Ai*∈

*σa*. The proof is

completed.

Now we consider the conditions that a family of*L*-*σ*-algebras forms anL, M-fuzzy

*σ*-algebra. ByTheorem 1.2, we can obtain the following result.

* Corollary 2.11. IfMis completely distributive, andσis an*L, M

*-fuzzyσ-algebra, then*

1*σ*_{}*b* ⊆*σafor anya, b*∈*M*\ {⊥*M*}*witha*∈*βb;*

2*σb*_{⊆}_{σ}a_{for any}_{a, b}_{∈}_{α}_{⊥}_{M}_{with}_{b}_{∈}_{αa}_{.}

**Theorem 2.12. Let***Mbe completely distributive, and let*{*σa* _{:} _{a}_{∈} _{α}_{⊥}_{M}_{}}_{be a family of}_{L}_{-}_{σ}_{}

*-algebras. Ifσa*_{}_{{}_{σ}b_{:}_{a}_{∈}_{αb}_{}}_{for all}_{a}_{∈}_{α}_{⊥}_{M}_{, then there exists an}_{L, M}_{-fuzzy}_{σ}_{-algebra}_{σ}

*such thatσa*_{}_{σ}a_{.}

*Proof. Suppose thatσa*_{}_{{}_{σ}b_{:}_{a}_{∈}_{αb}_{}}_{for all}_{a}_{∈}_{α}_{⊥}_{M}_{. Define}_{σ}_{:}_{L}X_{→} _{M}_{by}

*σA *

*a*∈*M*

a∨*σa*A {*a*∈*M*:*A*∈_{}*σa*}*.* _{}_{2.6}_{}

ByTheorem 1.2, we can obtain that*σaσa*.

**Corollary 2.13. Let***Mbe completely distributive, and let* {*σa* _{:} _{a}_{∈} _{α}_{⊥}_{M}_{}} _{be a family of}_{σ}_{}

*-algebras. Ifσa*_{}_{{}_{σ}b_{:}_{a}_{∈}_{αb}_{}}_{for all}_{a}_{∈}_{α}_{⊥}_{M}_{, then there exists an}_{M}_{-fuzzifying}_{σ}_{-algebra}_{σ}

*such thatσaσa.*

* Theorem 2.14. LetMbe completely distributive, and let*{

*σa*:

*a*∈

*M*\ {⊥

*M*}}

*be a family ofL-σ*

*-algebra. Ifσa*{*σb*:*b*∈*βa*}*for alla*∈*M*\ {⊥*M*}*, then there exists an*L, M*-fuzzyσ-algebra*
*σsuch thatσaσa.*

*Proof. Suppose thatσa*

{*σb*:*b*∈*βa*}for all*a*∈*M*\ {⊥*M*}. Define*σ*:*LX* → *M*by

*σA *

*a*∈*M*

a∧*σaA *{*a*∈*M*:*A*∈*σa*}*.* _{}_{2.7}_{}

ByTheorem 1.2, we can obtain*σ*_{}*aσa*.

* Corollary 2.15. LetMbe completely distributive, and let*{

*σa*:

*a*∈

*M*\ {⊥

*M*}}

*be a family ofσ*

*-algebra. Ifσa*{*σb*:*b*∈*βa*}*for alla*∈*M*\ {⊥*M*}*, then there exists anM-fuzzifyingσ-algebra*
*σsuch thatσ*_{}*aσa.*

* Theorem 2.16. Let* {

*σi*:

*i*∈ Ω}

*be a family of*L, M

*-fuzzy*

*σ-algebra on*

*X. Then*∧

*i*∈Ω*σi* *is an*

L, M*-fuzzyσ-algebra onX, where _{i}*

_{∈}

_{Ω}

*σi*:

*LX*

_{→}

_{M}_{is defined by}_{}

*i*∈Ω *σiA i*∈Ω *σiA.*

**3.**

*L, M*

**-Fuzzy Measurable Functions**

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

* Theorem 3.1. Let*Y, τ

*be an*L, M

*-fuzzy measurable space andf*:

*X*→

*Ya mapping. Define a*

*mappingf*←τ:

_{L}*LX*

_{→}

_{M}_{by for all}_{A}_{∈}

_{L}X_{,}*f _{L}*←τA

*τ*B:

*f*←B

_{L}*A,*

*where*∀

*x*∈

*X, f*←Bx

_{L}*B*

*f*x

*.*

_{}

_{3.1}

_{}

*Then*X, f* _{L}*←τ

*is an*L, M

*-fuzzy measurable space.*

*Proof.* **LMS1**holds from the following equality:

*f _{L}*←τ

*χ*∅

*τB*:

*fL*←B

*χ*∅

*τ* *χ*∅*M.* _{}_{3.2}_{}

**LMS2**can be shown from the following fact: for all*A*∈*LX*_{,}

*f _{L}*←τA

*τB*:

*f*←B

_{L}*A*

*τ* *B*:*f _{L}*←

*Bf*←B

_{L}*A*

*f _{L}*←τ

*A.*

3.3

**LMS3**for any{*An*:*n*∈N} ⊆*LX*, by

*f _{L}*←τ

*n*∈N

*An*

*τ*B:*f _{L}*←B

*n*∈N

*An*

≥

*τ*

_{}

*n*∈N

*Bn*

:*f _{L}*←Bn

*An*

≥

*n*∈N

*f _{L}*←τAn

3.4

we can proveLMS3.

*Definition 3.2. Let*X, σandY, τbeL, M-fuzzy measurable spaces. A mapping*f* :*X* →
*Y* is calledL, M-fuzzy measurable if*σf _{L}*←B≥

*τ*Bfor all

*B*∈

*LY*

_{.}

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.

* Theorem 3.3. Let*X, σ

*and*Y, τ

*be two*L, M

*-fuzzy measurable spaces. A mappingf*:

*X*→

*Y*

*is*L, M*-fuzzy measurable if and only iff _{L}*←τA≤

*σAfor allA*∈

*LX*

_{.}*Proof.*

*Necessity. Iff* : *X* → *Y* isL, M-fuzzy measurable, then*σf _{L}*←B ≥

*τ*Bfor all

*B*∈

*LY*

_{.}

Hence for all*B*∈*LY*_{, we have}

*f _{L}*←τA

*τB*:

*f*←B

_{L}*A*

≤*σ* *f _{L}*←B:

*f*←B

_{L}*A*

*σA.*

3.5

*Suﬃciency. Iff _{L}*←τA≤

*σA*for all

*A*∈

*LX*

_{, then}

_{τB}_{≤}

*←*

_{f}*L*τf*L*←B≤*σfL*←Bfor all
*B*∈*LY*_{; this shows that}_{f}_{:}_{X}_{→} _{Y}_{is}_{L, M}_{-fuzzy measurable.}

The next three theorems are trivial.

* Theorem 3.4. Iff* :X, σ → Y, τ

*andf*:Y, τ → Z, ρ

*are*L, M

*-fuzzy measurable, then*

*g*◦*f* :X, σ → Z, ρ*is*L, M*-fuzzy measurable.*

* Theorem 3.5. Let* X, σ

*and*Y, τ

*be*L, M

*-fuzzy measurable spaces. Then a mapping*

*f*:

X, σ → Y, τ*is*L, M*-fuzzy measurable if and only iff*:X, σ*a* → Y, τ*aisL-measurable*

*for anya*∈*M*\ {⊥*M*}*.*

**Theorem 3.6. Let***M* *be completely distributive, and let* X, σ *and* Y, τ *be* L, M*-fuzzy*
*measurable spaces. Then a mappingf* : X, σ → Y, τ *is*L, M*-fuzzy measurable if and only*
*iff*:X, σ*a*_{} _{→} _{Y, τ}*a*_{}_{is}_{L}_{-measurable for any}_{a}_{∈}_{α}_{⊥}_{M}_{.}

* Corollary 3.7. Let* X, σ

*and*Y, τ

*be*

*M-fuzzifying measurable spaces. Then a mapping*

*f*:

X, σ → Y, τ*isM-fuzzifying measurable if and only iff* :X, σ*a* → Y, τ*ais measurable*

*for anya*∈*M*\ {⊥*M*}*.*

**Corollary 3.8. Let***M* *be completely distributive, and let* X, σ *and* Y, τ *be* *M-fuzzifying*
*measurable spaces. Then a mapping* *f* : X, σ → Y, τ*isM-fuzzifying measurable if and only*
*iff*:X, σ*a*_{} _{→} _{Y, τ}*a*_{}_{is measurable for any}_{a}_{∈}_{α}_{⊥}_{M}_{.}

**4.**

*I, I*

**-Fuzzy**

*σ*

**-Algebras Generated by**

*I*

**-Fuzzifying**

*σ*

**-Algebras**

In this section,Bwill be used to denote the*σ*-algebra of Borel subsets of*I* 0*,*1.

* Theorem 4.1. Let*X, σ

*be anI-fuzzifying measurable space. Define a mappingζσ*:

*IX*

_{→}

_{I}_{by}*ζσA *

*B*∈B

*Thenζσis a stratified*I, I*-fuzzyσ-algebra, which is said to be the*I, I*-fuzzyσ-algebra generated*
*byσ.*

*Proof.* **LMS1**For any*B*∈ Band for any*a*∈*I*, if*a*∈*B*, thena∧*χX*−1B *X*; if*a*∈_{}*B*, then

a∧*χX*−1B ∅. However, we have that*σa*∧*χX*−1B 1. This shows that*ζσa*∧*χX*

1.

**LMS2**for all*A*∈*IX*_{and for all}_{B}_{∈ B}_{, we have}

*ζσ* *A*
*B*∈B

*σ*1−*A*−1B

*B*∈B

*σ*{*x*∈*X*: 1−*Ax*∈*B*}

*B*∈B

*σ*{*x*∈*X*:∃*b*∈*B,* s*.*t*. Ax *1−*b*}

*B*∈B

*σA*−1B

*ζσA.*

4.2

**LMS3**for any{*An*:*n*∈N} ⊆*LX*and for all *B*∈ B, by

*ζσ*

*n*∈N

*An*
*B*∈B
*σ*
⎛
⎝

*n*∈N

*An*
_{−}1
B
⎞
⎠
*B*∈B
*σ*
_{}

*n*∈N

*A*−* _{n}*1B

≥

*B*∈B

*n*∈N

*σA*−* _{n}*1B

*n*∈N

*B*∈B

*σA*−* _{n}*1B

*n*∈N

*ζσAn,*

4.3

we obtain*ζσ _{n}*

_{∈}

_{N}

*An*≥

_{n}_{∈}

_{N}

*ζσAn*.

* Corollary 4.2. Let* X, σ

*be a measurable space. Define a subset*

*ζσ*⊆

*IX*

_{}

_{can be viewed as a}*mappingζσ*:*IX* _{→} _{2}_{}_{by}

*ζσ A*∈*IX*:∀*B*∈ B*, A*−1B∈*σ.* 4.4

*Thenζσis a stratifiedI-σ-algebra.*

* Theorem 4.3. Let*X, σ

*and*Y, τ

*be twoI-fuzzifying measurable spaces, andf*:

*X*→

*Y*

*is a*

*map. Thenf*:X, σ → Y, τ

*isI-fuzzifying measurable if and only iff*:X, ζσ → Y, ζτ

*is*

I, I*-fuzzy measurable.*

*Proof.*

*Necessity. Suppose thatf* : X, σ → Y, τis*I*-fuzzifying measurable. Then *σf*−1_{A} _{≥}

*τA* for any *A* ∈ **2***X*_{. In order to prove that} _{f}_{:} _{X, ζσ} _{→} _{Y, ζτ} _{is} _{I, I}_{}_{-fuzzy}

measurable, we need to prove that*ζσf _{L}*←A≥

*ζτA*for any

*A*∈

*IX*

_{.}

In fact, for any*A*∈*IX*_{, by}

*ζσ* *f _{L}*←A

*B*∈B

*σ* *f _{L}*←A−1B

*B*∈B

*σ* *A*◦*f*−1B

*B*∈B

*σ* *B*◦*A*◦*f*
*B*∈B

*σf*−1*A*−1B

≥

*B*∈B

*τA*−1B *ζτA,*

4.5

we can prove the necessity.

*Suﬃciency. Suppose that* *f* : X, ζσ → Y, ζτ is I, I-fuzzy measurable. Then

*ζσf*←

*I* A≥ *ζτA*for any*A*∈*IX*. In particular, it follows that*ζσfI*←A≥*ζτ*A

for any*A*∈**2***X*. In order to prove that*f* :X, σ → Y, τis*I*-fuzzifying measurable, we need
to prove that*σf*−1_{A} _{≥} _{τA}_{for any}_{A}_{∈} _{2}*X*_{. In fact, for any}_{A}_{∈} _{2}*X* _{and for any}_{B}_{∈ B}_{,}

if 0*,*1 ∈ *B*, then*A*−1_{B } _{X}_{; if 0}_{,}_{1}_{∈}_{}_{B}_{, then}* _{A}*−1

_{B }

_{∅}

_{; if only one of 0 and 1 is in}

_{B}_{, then}

*A*−1B *A*or*A*−1B *A*. However, we have

*σ* *f _{I}*←A

*σ*

*f*←A

_{I}*σ* *f _{I}*←A∧

*σ*

*f*←A

_{I}

*B*∈B

*σ* *f _{L}*←A−1B

*ζσ* *f _{L}*←A

≥*ζτA*

*ζτA*∧*ζτ* *A*

*B*∈B

*τA*−1B*τA.*

4.6

This shows that*f* :X, σ → Y, τis*I*-fuzzifying measurable.

* Corollary 4.4. Let*X, σ

*and*Y, τ

*be two measurable spaces, andf*:

*X*→

*Y*

*is a mapping. Then*

**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 China10971242.

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