IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi Publishing Corporation35658110.1155/2010/356581356581Research Article(L,M)-Fuzzy σ-AlgebrasShiFu-Gui1VolodinAndreiDepartment of MathematicsSchool of ScienceBeijing Institute of TechnologyBeijing 100081Chinabit.edu.cn2010242201020102308200919122009190120102010Copyright © 2010This 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 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  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 .

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:LXM 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'. X is a nonempty set. LX 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 χA. The smallest element and the largest element in M are denoted by M and M, respectively.

The binary relation in M is defined as follows: for a,bM, ab if and only if for every subset DM, the relation bsupD always implies the existence of dD with ad . {aM:ab} is called the greatest minimal family of b in the sense of , denoted by β(b). Moreover, for bM, we define α(b)={aM:aopb}. In a completely distributive lattice M, there exist α(b) and β(b) for each bM, and b=β(b)=α(b) (see ).

In , 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, ALX and aL, we use the following notation:

A[a]={xX:A(x)a}.

If L is completely distributive, then we can define

A[a]={xX:aα(A(x))}.

Some properties of these cut sets can be found in .

Theorem 1.1 (see [<xref ref-type="bibr" rid="B12">4</xref>]).

Let M be a completely distributive lattice and {ai:iΩ}M. Then

α(iΩai)=iΩα(ai), that is, α is an - map;

β(iΩai)=iΩβ(ai), that is, β is a union-preserving map.

For aL and DX, we define two L-fuzzy sets aD and aD as follows: (aD)(x)={a,xD;0,xD.(aD)(x)={1,xD;a,xD. Then for each L-fuzzy set A in LX, it follows that A=aL(aA[a]).

Theorem 1.2 (see [<xref ref-type="bibr" rid="B3">5</xref>, <xref ref-type="bibr" rid="B8">7</xref>, <xref ref-type="bibr" rid="B11">10</xref>]).

If L is completely distributive, then for each L-fuzzy set A in LX, we have

A=aL(aA[a])=aL(aA[a]);

for  all  aL, A[a]=bβ(a)A[b];

for  all  aL, A[a]=aα(b)A[b].

For a family of L-fuzzy sets {Ai:iΩ} in LX, it is easy to see that (iΩAi)[a]=iΩ(Ai)[a]. If L is completely distributive, then it follows  that (iΩAi)[a]=iΩ(Ai)[a].

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:

for any constant fuzzy set α, ασ;

for any A[0,1]X, 1-Aσ;

for any {An:n}σ, n  Anσ.

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 and X a nonempty set. A subset σ of LX is called an L-σ-algebra if it satisfies the following three conditions:

for any aL, constant L-fuzzy set aχXσ;

for any ALX, Aσ;

for any {An:n}σ, n  Anσ.

The L-fuzzy sets in σ are called L-measurable sets, and the pair (X,σ) an L-measurable space.

2. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M156"><mml:mo stretchy="false">(</mml:mo><mml:mi>L</mml:mi><mml:mo>,</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>-Fuzzy <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M157"><mml:mrow><mml:mi>σ</mml:mi></mml:mrow></mml:math></inline-formula>-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.

Let X be a nonempty set. A mapping σ:LXM is called an (L,M)-fuzzy σ-algebra if it satisfies the following three conditions:

σ(χ)=M;

for any ALX, σ(A)=σ(A);

for any {An:n}LX, σ(n  An)n  σ(An).

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

(LMS1)*   aL, σ(aχX)=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 LX is regarded as a mapping σ:LX2, then σ is an L-σ-algebra if and only if it satisfies the following conditions:

χσ;

AσAσ;

for any {An:n}σ, n  Anσ.

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:2XM is an M-fuzzifying σ-algebra if and only if it satisfies the following conditions:

σ()=M;

for any A2X, σ(A)=σ(A);

for any {An:n}2X, σ(n  An)n  σ(An).

Example 2.3.

Let  (X,σ) be a crisp measurable space. Defineχσ:2X[0,1] by χσ(A)={1,Aσ;0,Aσ. Then it is easy to prove that (X,χσ) is a [0,1]-fuzzifying measurable space.

Example 2.4.

Let X be a nonempty set and σ:2X[0,1] a mapping defined by σ(A)={1,A{,X};0.5,A{,X}. Then it is easy to prove that (X,σ) is a [0,1]-fuzzifying measurable space. If A2X with A{,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[0,1] a mapping defined by σ(A)={1,A{χ,χX};0.5,A{χ,χX}. 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{χ,χX}, 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 {An:n}LX, σ(n  An)n  σ(An).

Proof.

This can be proved from the following fact: σ(nAn)=σ(n(An))nσ((An))=nσ(An). The next two theorems give characterizations of an (L,M)-fuzzy σ-algebra.

Theorem 2.7.

A mapping σ:LXM is an (L,M)-fuzzy σ-algebra if and only if for each aM{M}, σ[a] is an L-σ-algebra.

Proof.

The proof is obvious and is omitted.

Corollary 2.8.

A mapping σ:2XM is an M-fuzzifying σ-algebra if and only if for each aM{M}, σ[a] is a σ-algebra.

Theorem 2.9.

If M is completely distributive, then a mapping σ:LXM is an (L,M)-fuzzy σ-algebra if and only if for each aα(M), σ[a] is an L-σ-algebra.

Proof.

Necessity..

Suppose that σ:LXM is an (L,M)-fuzzy σ-algebra and aα(M). Now we prove that σ[a] is an L-σ-algebra.

By σ(χ)=M and α(M)=, we know that aα(σ(χ)); this implies that χσ[a].

If Aσ[a], then aα(σ(A))=α(σ(A)); this shows that Aσ[a].

If {Ai:iΩ}σ[a], then for  all  iΩ, aα(σ(Ai)). Hence aiΩα(σ(Ai)). By σ(iΩ  Ai)iΩσ(Ai), we know that

α(σ(iΩAi))α(iΩσ(Ai))=iΩα(σ(Ai)). This shows that aα(σ(iΩ  Ai)). Therefore, iΩ  Aiσ[a]. The proof is completed.

Corollary 2.10.

If M is completely distributive, then a mapping σ:2XM is an M-fuzzifying σ-algebra if and only if for each aα(M), σ[a] 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

σ[b]σ[a] for any a,bM{M} with aβ(b);

σ[b]σ[a] for any a,bα(M) with bα(a).

Theorem 2.12.

Let M be completely distributive, and let {σa:    aα(M)} be a family of L-σ-algebras. If σa={σb:aα(b)} for all aα(M), then there exists an (L,M)-fuzzy σ-algebra σ such that σ[a]=σa.

Proof.

Suppose that σa={σb:aα(b)} for all aα(M). Define σ:LXM by σ(A)=aM(aσa(A))={aM:Aσa}. By Theorem 1.2, we can obtain that σ[a]=σa.

Corollary 2.13.

Let M be completely distributive, and let {σa:  aα(M)} be a family of σ-algebras. If σa={σb:aα(b)} for all aα(M), then there exists an M-fuzzifying σ-algebra σ such that σ[a]=σa.

Theorem 2.14.

Let M be completely distributive, and let {σa:  aM{M}} be a family of L-σ-algebra. If σa={σb:bβ(a)} for all aM{M}, then there exists an (L,M)-fuzzy σ-algebra σ such that σ[a]=σa.

Proof.

Suppose that σa={σb:bβ(a)} for all aM{M}. Define σ:LXM by σ(A)=aM(aσa(A))={aM:Aσa}. By Theorem 1.2, we can obtain σ[a]=σa.

Corollary 2.15.

Let M be completely distributive, and let {σa:  aM{M}} be a family of σ-algebra. If σa={σb:bβ(a)} for all aM{M}, then there exists an M-fuzzifying σ-algebra σ such that σ[a]=σa.

Theorem 2.16.

Let {σi:iΩ} be a family of (L,M)-fuzzy σ-algebra on X. Then iΩσi is an (L,M)-fuzzy σ-algebra on X, where iΩ  σi:LXM is defined by (iΩ  σi)(A)=iΩ  σi(A).

Proof.

This is straightforward.

3. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M380"><mml:mo stretchy="false">(</mml:mo><mml:mi>L</mml:mi><mml:mo>,</mml:mo><mml:mi>M</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>-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 and f:XY a mapping. Define a mapping fL(τ):LXM by for  all  ALX, fL(τ)(A)={τ(B):fL(B)=A},wherexX,  fL(B)(x)=B(f(x)). Then (X,fL(τ)) is an (L,M)-fuzzy measurable space.

Proof.

(LMS1) holds from the following equality: fL(τ)(χ)={τ(B):fL(B)=χ}=τ(χ)=M.

(LMS2) can be shown from the following fact: for  all  ALX, fL(τ)(A)={τ(B):fL(B)=A}={τ(B):fL(B)=fL(B)=A}=fL(τ)(A).

(LMS3) for any {An:n}LX, byfL(τ)(nAn)={τ(B):fL(B)=nAn}{τ(nBn):fL(Bn)=An}nfL(τ)(An)we  can  prove  (LMS3).

Definition 3.2.

Let (X,σ) and (Y,τ) be (L,M)-fuzzy measurable spaces. A mapping f:XY is called (L,M)-fuzzy measurable if σ(fL(B))τ(B) for all BLY.

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 (Y,τ) be two (L,M)-fuzzy measurable spaces. A mapping f:XY is (L,M)-fuzzy measurable if and only if fL(τ)(A)σ(A) for all ALX.

Proof.

Necessity..

If f:XY is (L,M)-fuzzy measurable, then σ(fL(B))τ(B) for all BLY. Hence for all BLY, we have

fL(τ)(A)={τ(B):fL(B)=A}{σ(fL(B)):fL(B)=A}=σ(A).Sufficiency..

If fL(τ)(A)σ(A) for all ALX, then τ(B)fL(τ)(fL(B))σ(fL(B)) for all BLY; this shows that f:XY is (L,M)-fuzzy measurable.

The next three theorems are trivial.

Theorem 3.4.

If f:(X,σ)(Y,τ) and f:(Y,τ)(Z,ρ) are (L,M)-fuzzy measurable, then gf:(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 if f:(X,σ[a])(Y,τ[a]) is L-measurable for any aM{M}.

Theorem 3.6.

Let M be completely distributive, and 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 if f:(X,σ[a])(Y,τ[a]) is L-measurable for any aα(M).

Corollary 3.7.

Let (X,σ) and (Y,τ) be M-fuzzifying measurable spaces. Then a mapping f:(X,σ)(Y,τ) is M-fuzzifying measurable if and only if f:(X,σ[a])(Y,τ[a]) is measurable for any aM{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) is M-fuzzifying measurable if and only if f:(X,σ[a])(Y,τ[a]) is measurable for any aα(M).

4. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M464"><mml:mo stretchy="false">(</mml:mo><mml:mi>I</mml:mi><mml:mo>,</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>-Fuzzy <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M465"><mml:mrow><mml:mi>σ</mml:mi></mml:mrow></mml:math></inline-formula>-Algebras Generated by <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M466"><mml:mrow><mml:mi>I</mml:mi></mml:mrow></mml:math></inline-formula>-Fuzzifying <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M467"><mml:mrow><mml:mi>σ</mml:mi></mml:mrow></mml:math></inline-formula>-Algebras

In this section, 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):IXI by ζ(σ)(A)=Bσ(A-1(B)). 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 B and for any aI, if aB, then (aχX)-1(B)=X; if aB, then (aχX)-1(B)=. However, we have that σ((aχX)-1(B))=1. This shows that ζ(σ)(aχX)=1.

(LMS2)for  all  AIX and for  all  B, we have ζ(σ)(A)=Bσ((1-A)-1(B))=Bσ({xX:1-A(x)B})=Bσ({xX:bB,  s.t.    A(x)=1-b})=Bσ(A-1(B))=ζ(σ)(A).

(LMS3) for any {An:n}LX and for  all  B, by ζ(σ)(nAn)=Bσ((nAn)-1(B))=Bσ(nAn-1(B))Bnσ(An-1(B))=nBσ(An-1(B))=nζ(σ)(An),

we obtain ζ(σ)(nAn)nζ(σ)(An).

Corollary 4.2.

Let (X,σ) be a measurable space. Define a subset ζ(σ)IX(can be viewed as a mapping ζ(σ):IX2) by ζ(σ)={AIX:B,A-1(B)σ}. 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 (Y,τ) be two I-fuzzifying measurable spaces, and f:XY is a map. Then f:(X,σ)(Y,τ) is I-fuzzifying measurable if and only if f:(X,ζ(σ))(Y,ζ(τ)) is (I,I)-fuzzy measurable.

Proof.

Necessity.

Suppose that f:(X,σ)(Y,τ) is I-fuzzifying measurable. Then σ(f-1(A))τ(A) for any A2X. In order to prove that f:(X,ζ(σ))(Y,ζ(τ)) is (I,I)-fuzzy measurable, we need to prove that ζ(σ)(fL(A))ζ(τ)(A) for any AIX.

In fact, for any AIX, by ζ(σ)(fL(A))  =Bσ((fL(A))-1(B))  =Bσ((Af)-1(B))=Bσ(BAf)  =Bσ(f-1(A-1(B)))Bτ(A-1(B))  =ζ(τ)(A), we can prove the necessity.

Sufficiency.

Suppose that f:(X,ζ(σ))(Y,ζ(τ)) is (I,I)-fuzzy measurable. Then ζ(σ)(fI(A))ζ(τ)(A) for any AIX. In particular, it follows that ζ(σ)(fI(A))ζ(τ)(A) for any A2X. In order to prove that f:(X,σ)(Y,τ) is I-fuzzifying measurable, we need to prove that σ(f-1(A))τ(A) for any A2X. In fact, for any A2X and for any B, if 0,1B, then A-1(B)=X; if 0,1B, then A-1(B)=; if only one of 0 and 1 is in B, then A-1(B)=A or A-1(B)=A'. However, we have

σ(fI(A))=σ(fI(A))=σ(fI(A))σ(fI(A))=Bσ((fL(A))-1(B))=ζ(σ)(fL(A))ζ(τ)(A)=ζ(τ)(A)ζ(τ)(A)=Bτ(A-1(B))=τ(A).

This shows that f:(X,σ)(Y,τ) is I-fuzzifying measurable.

Corollary 4.4.

Let (X,σ) and (Y,τ) be two measurable spaces, and f:XY is a mapping. Then f:(X,σ)(Y,τ) is measurable if and only if f:(X,ζ(σ))(Y,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).

KlementE. P.Fuzzy σ-algebras and fuzzy measurable functionsFuzzy Sets and Systems1980418393MR58083610.1016/0165-0114(80)90066-4BiacinoL.LettieriA.L-σ-algebras and L-measuresFuzzy Sets and Systems199144221922510.1016/0165-0114(91)90005-BMR1140856 (93b:28050)DwingerPh.Characterization of the complete homomorphic images of a completely distributive complete lattice. IIndagationes Mathematicae1982444403414MR683528ZBL0503.06012WangG.-J.Theory of topological molecular latticesFuzzy Sets and Systems199247335137610.1016/0165-0114(92)90308-QMR1166284ZBL0783.54032HuangH.-L.ShiF.-G.L-fuzzy numbers and their propertiesInformation Sciences200817841141115110.1016/j.ins.2007.10.001MR2369556ZBL1136.03326NegoiţăC. V.RalescuD. A.Applications of Fuzzy Sets to Systems Analysis197511New York, NY, USABirkhäuser, Basel, Switzerland; Stuttgart and Halsted Press191Interdisciplinary Systems Research SeriesMR0490082ShiF.-G.The theory and applications of Lβ-nested sets and Lα-nested sets and its applicationsFuzzy Systems and Mathematics1995946572MR1384670ShiF.-G.L-fuzzy sets and prime element nested setsJournal of Mathematical Research and Exposition1996163398402MR1408582ZBL0899.04004ShiF.-G.Theory of molecular nested sets and its applicationsYantai Normal University Journal199613336ShiF.-G.L-fuzzy relations and L-fuzzy subgroupsJournal of Fuzzy Mathematics200082491499MR1767444