(L,M) Fuzzy σ Algebras

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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 ofL, M-fuzzyσ-algebras is introduced in the lattice value fuzzy set theory. It is a generalization of Klement’s fuzzyσ-algebras. In our definition ofL, M-fuzzyσ-algebras, each

L-fuzzy subset can be regarded as anL-measurable set to some degree.

1. Introduction and Preliminaries

In 1980, Klement established an axiomatic theory of fuzzyσ-algebras in1in order to prepare a measure theory for fuzzy sets. In the definition of Klement’s fuzzyσ-algebraX, σ,σwas defined as a crisp family of fuzzy subsets of a setXsatisfying certain set of axioms. In 1991, Biacino and Lettieri generalized Klement’s fuzzyσ-algebras toL-fuzzy setting2.

In this paper, when bothL andMare complete lattices, we define anL, M-fuzzy

σ-algebra on a nonempty setXby means of a mappingσ:LX Msatisfying three axioms.

Thus eachL-fuzzy subset ofXcan be regarded as anL-measurable set to some degree. Whenσis anL, M-fuzzyσ-algebra onX,X, σis called anL, M-fuzzy measurable space. AnL,2-fuzzyσ-algebra is also called an L-σ-algebra. A Klementσ-algebra can be viewed as a stratified 0,1-σ-algebra. A Biacino-Lettieri L-σ-algebra can be viewed as a stratifiedL-σ-algebra. A2, M-fuzzyσ-algebra is also called anM-fuzzifyingσ-algebra. A crispσ-algebra can be regarded as a2,2-fuzzyσ-algebra.

Throughout this paper, bothLandMdenote complete lattices, andL has an order-reversing involution’. X is a nonempty set. LX is the set of all L-fuzzy setsor L-sets for

shortonX. We often do not distinguish a crisp subsetAofXand its character functionχA. The smallest element and the largest element inMare denoted by⊥MandM, respectively.

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βb. Moreover, forbM, we defineαb {aM : aopb}. In a completely distributive

latticeM, there existαbandβbfor eachbM, andbβb αb see4.

In4, Wang thought thatβ0 {0}andα1 {1}. In fact, it should be thatβ0 ∅ andα1 ∅.

For a complete latticeL,ALXandaL, we use the following notation:

Aa{xX :Axa}. 1.1

IfLis completely distributive, then we can define

Aa{xX:a

αAx}. 1.2

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

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

1αiΩai iΩαai, that is,αis anmap;

2βiΩai iΩβai, that is,βis a union-preserving map. ForaLandDX, we define twoL-fuzzy setsaDandaDas follows:

a∧Dx

⎧ ⎨ ⎩

a, xD;

0, xD. a∨Dx

⎧ ⎨ ⎩

1, xD;

a, xD. 1.3

Then for eachL-fuzzy setAinLX, it follows that

A

aL

aAa. 1.4

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

have

1AaLa∧Aa

aLa∨Aa; 2for allaL,Aa

bβaAb; 3for allaL,Aa

aαbAb.

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

i∈Ω

Ai

a

i∈Ω

Aia. 1.5

IfLis completely distributive, then it follows [7] that

i∈Ω

Ai

a

i∈Ω

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

1for any constant fuzzy setα,ασ;

2for anyA∈0,1X, 1−Aσ;

3for any{An:n∈N} ⊆σ,

n∈NAnσ.

The fuzzy sets inσare 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 anyaL, constantL-fuzzy setaχXσ; 2for anyALX,Aσ;

3for any{An:n∈N} ⊆σ,n∈NAnσ.

TheL-fuzzy sets inσare calledL-measurable sets, and the pairX, σanL-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 anM-fuzzy subsetσofLX.

Definition 2.1. LetX be a nonempty set. A mappingσ : LX Mis called anL, M-fuzzy σ-algebra if it satisfies the following three conditions:

LMS1σχM;

LMS2for anyALX,σA σA;

LMS3for any{An:n∈N} ⊆LX,σ

n∈N Ann∈N σAn.

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

LMS1aL,σaχX M.

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

AnL,2-fuzzyσ-algebra is also called anL-σ-algebra, and anL,2-fuzzy measurable space is also called anL-measurable space.

A2, M-fuzzyσ-algebra is also called anM-fuzzifyingσ-algebra, and a2, M-fuzzy measurable space is also called anM-fuzzifying measurable space.

Obviously a crisp measurable space can be regarded as a 2,2-fuzzy measurable space.

Ifσis anL, M-fuzzyσ-algebra, thenσAcan be regarded as the degree to whichA

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Remark 2.2. If a subsetσofLXis regarded as a mappingσ:LX 2, thenσis anL-σ-algebra

if and only if it satisfies the following conditions:

LS1χ∅ ∈σ;

LS2AσAσ;

LS3for any{An:n∈N} ⊆σ,nNAnσ.

Thus we easily see that a Klement σ-algebra is exactly a stratified 0,1-σ-algebra, and a Biacino-LettieriL-σ-algebra is exactly a stratifiedL-σ-algebra.

Moreover, whenL 2, a mappingσ :2X Mis anM-fuzzifyingσ-algebra if and

only if it satisfies the following conditions:

MS1σM;

MS2for anyA2X,σA σA;

MS3for any{An:n∈N} ⊆2X,σ

n∈NAn

n∈N σAn.

Example 2.3. LetX, σbe a crisp measurable space. Defineχσ :2X → 0,1by

χσA

⎧ ⎨ ⎩

1, Aσ;

0, Aσ.

2.1

Then it is easy to prove thatX, χσis a0,1-fuzzifying measurable space.

Example 2.4. LetXbe a nonempty set andσ:2X 0,1a 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. IfA2X with A∈{∅ , X}, then 0.5 is the degree to whichAis measurable.

Example 2.5. LetXbe a nonempty set andσ:0,1X → 0,1a mapping defined by

σA

⎧ ⎨ ⎩

1, Aχ, χX;

0.5, Aχ, χX.

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Then it is easy to prove thatX, σis a0,1,0,1-fuzzy measurable space. IfA ∈0,1X withA∈{ χ, χX}, then 0.5 is the degree to whichAis0,1-measurable.

Proposition 2.6. LetX, σbe anL, M-fuzzy measurable spaces. Then for any{An:n∈N} ⊆LX, σn∈NAnn∈NσAn.

Proof. This can be proved from the following fact:

σ

n∈N An

σ

n∈N An

n∈N

σ An n∈N

σAn. 2.4

The next two theorems give characterizations of anL, M-fuzzyσ-algebra.

Theorem 2.7. A mappingσ : LX Mis anL, M-fuzzyσ-algebra if and only if for eacha M\ {⊥M},σais anL-σ-algebra.

Proof. The proof is obvious and is omitted.

Corollary 2.8. A mappingσ : 2X Mis anM-fuzzifyingσ-algebra if and only if for each a M\ {⊥M},σais aσ-algebra.

Theorem 2.9. IfMis completely distributive, then a mappingσ :LX Mis anL, M-fuzzy σ-algebra if and only if for eachaαM,σais anL-σ-algebra.

Proof.

Necessity. Suppose thatσ :LX Mis anL, M-fuzzyσ-algebra andaαM. Now we

prove thatσais anL-σ-algebra.

LS1Byσχ MandαM ∅, we know thataασχ∅; this implies thatχ∅∈σa.

LS2IfAσa, thenaασA ασA; this shows thatAσa.

LS3If{Ai : i ∈Ω} ⊆ σa, then for alli ∈ Ω,aασAi. Hencea

i∈ΩασAi. By

σi∈Ω Aii∈ΩσAi, we know that

α

σ

i∈Ω

Ai

α

i∈Ω

σAi

i∈Ω

ασAi. 2.5

This shows that aασiΩ Ai. Therefore, iΩ Aiσa. The proof is

completed.

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

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

Corollary 2.11. IfMis completely distributive, andσis anL, M-fuzzyσ-algebra, then

1σbσafor anya, bM\ {⊥M}withaβb;

2σbσafor anya, bαMwithbαa.

Theorem 2.12. Let Mbe completely distributive, and let{σa : a αM}be a family of L-σ

-algebras. Ifσa{σb :aαb}for allaαM, then there exists anL, M-fuzzyσ-algebraσ

such thatσaσa.

Proof. Suppose thatσa{σb:aαb}for allaαM. Defineσ:LX Mby

σA

aM

a∨σaA {aM:Aσa}. 2.6

ByTheorem 1.2, we can obtain thatσaσa.

Corollary 2.13. Let Mbe completely distributive, and let {σa : a αM} be a family ofσ

-algebras. Ifσa{σb :aαb}for allaαM, then there exists anM-fuzzifyingσ-algebraσ

such thatσaσa.

Theorem 2.14. LetMbe completely distributive, and let{σa: aM\ {⊥M}}be a family ofL-σ

-algebra. Ifσa{σb:bβa}for allaM\ {⊥M}, then there exists anL, M-fuzzyσ-algebra σsuch thatσaσa.

Proof. Suppose thatσa

{σb:bβa}for allaM\ {⊥M}. Defineσ:LXMby

σA

aM

a∧σaA {aM:Aσa}. 2.7

ByTheorem 1.2, we can obtainσaσa.

Corollary 2.15. LetMbe completely distributive, and let{σa : aM\ {⊥M}}be a family ofσ

-algebra. Ifσa{σb:bβa}for allaM\ {⊥M}, then there exists anM-fuzzifyingσ-algebra σsuch thatσaσa.

Theorem 2.16. Let {σi : i ∈ Ω} be a family of L, M-fuzzy σ-algebra on X. Then

i∈Ωσi is an

L, M-fuzzyσ-algebra onX, whereiΩ σi:LX Mis defined by

i∈Ω σiA i∈Ω σiA.

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3.

L, M

-Fuzzy Measurable Functions

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

Theorem 3.1. LetY, τbe anL, M-fuzzy measurable space andf :XYa mapping. Define a mappingfL←τ:LX Mby for allALX,

fL←τA τB:fL←B A, wherexX, fL←Bx B fx. 3.1

ThenX, fL←τis anL, M-fuzzy measurable space.

Proof. LMS1holds from the following equality:

fL←τ χτB:fL←B χ

τ χM. 3.2

LMS2can be shown from the following fact: for allALX,

fL←τA τB:fL←B A

τ B:fLBfL←BA

fL←τ A.

3.3

LMS3for any{An:n∈N} ⊆LX, by

fL←τ

n∈N

An

τB:fL←B n∈N

An

τ

n∈N

Bn

:fL←Bn An

n∈N

fL←τAn

3.4

we can proveLMS3.

Definition 3.2. LetX, σandY, τbeL, M-fuzzy measurable spaces. A mappingf :XY is calledL, M-fuzzy measurable ifσfL←B≥τBfor allBLY.

An L,2-fuzzy measurable mapping is called an L-measurable mapping, and a

2, M-fuzzy measurable mapping is called anM-fuzzifying measurable mapping.

Obviously a Klement fuzzy measurable mapping can be viewed as an 0,1 -measurable mapping.

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Theorem 3.3. LetX, σandY, τbe twoL, M-fuzzy measurable spaces. A mappingf:XY

isL, M-fuzzy measurable if and only iffL←τA≤σAfor allALX.

Proof.

Necessity. Iff : XY isL, M-fuzzy measurable, thenσfL←B ≥ τBfor allBLY.

Hence for allBLY, we have

fL←τA τB:fL←B A

σ fL←B:fL←B A

σA.

3.5

Sufficiency. IffL←τA≤σAfor allALX, thenτBf

LτfL←B≤σfL←Bfor all BLY; this shows thatf:X Y isL, M-fuzzy measurable.

The next three theorems are trivial.

Theorem 3.4. Iff :X, σ → Y, τandf :Y, τ → Z, ρareL, M-fuzzy measurable, then

gf :X, σ → Z, ρisL, 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, τisL, M-fuzzy measurable if and only iff:X, σa → Y, τaisL-measurable

for anyaM\ {⊥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 isL, M-fuzzy measurable if and only iff:X, σa Y, τaisL-measurable for anyaαM.

Corollary 3.7. Let X, σ and Y, τ be M-fuzzifying measurable spaces. Then a mapping f :

X, σ → Y, τisM-fuzzifying measurable if and only iff :X, σa → Y, τais measurable

for anyaM\ {⊥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, τisM-fuzzifying measurable if and only iff:X, σa Y, τais measurable for anyaαM.

4.

I, I

-Fuzzy

σ

-Algebras Generated by

I

-Fuzzifying

σ

-Algebras

In this section,Bwill be used to denote theσ-algebra of Borel subsets ofI 0,1.

Theorem 4.1. LetX, σbe anI-fuzzifying measurable space. Define a mappingζσ:IX Iby

ζσA

B∈B

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Thenζσis a stratifiedI, I-fuzzyσ-algebra, which is said to be theI, I-fuzzyσ-algebra generated byσ.

Proof. LMS1For anyB∈ Band for anyaI, ifaB, thena∧χX−1B X; ifaB, then

a∧χX−1B ∅. However, we have thatσaχX−1B 1. This shows thatζσaχX

1.

LMS2for allAIXand for allB∈ B, we have

ζσ A B∈B

σ1−A−1B

B∈B

σ{xX: 1−AxB}

B∈B

σ{xX:∃bB, s.t. Ax 1−b}

B∈B

σA−1B

ζσA.

4.2

LMS3for any{An:n∈N} ⊆LXand for all B∈ B, by

ζσ

n∈N

An B∈B σ ⎛ ⎝

n∈N

An 1 B ⎞ ⎠ B∈B σ

n∈N

An1B

B∈B

n∈N

σAn1B

n∈N

B∈B

σAn1B n∈N

ζσAn,

4.3

we obtainζσnNAnnNζσAn.

Corollary 4.2. Let X, σbe a measurable space. Define a subset ζσIXcan be viewed as a

mappingζσ:IX 2by

ζσ AIX:∀B∈ B, A−1B∈σ. 4.4

Thenζσis a stratifiedI-σ-algebra.

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Theorem 4.3. LetX, σandY, τbe twoI-fuzzifying measurable spaces, andf : XY is a map. Thenf :X, σ → Y, τisI-fuzzifying measurable if and only iff:X, ζσ → Y, ζτis

I, I-fuzzy measurable.

Proof.

Necessity. Suppose thatf : X, σ → Y, τisI-fuzzifying measurable. Then σf−1A

τA for any A2X. In order to prove that f : X, ζσ Y, ζτ is I, I-fuzzy

measurable, we need to prove thatζσfL←A≥ζτAfor anyAIX.

In fact, for anyAIX, by

ζσ fL←A B∈B

σ fL←A−1B B∈B

σ Af−1B

B∈B

σ BAf B∈B

σf−1A−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≥ ζτAfor anyAIX. In particular, it follows thatζσfI←A≥ζτA

for anyA2X. In order to prove thatf :X, σ → Y, τisI-fuzzifying measurable, we need to prove thatσf−1A τAfor anyA 2X. In fact, for anyA 2X and for anyB∈ B,

if 0,1 ∈ B, thenA−1B X; if 0,1B, thenA−1B ; if only one of 0 and 1 is inB, then

A−1B AorA−1B A. However, we have

σ fI←Aσ fI←A

σ fI←A∧σ fI←A

B∈B

σ fL←A−1B

ζσ fL←A

ζτA

ζτAζτ A

B∈B

τA−1BτA.

4.6

This shows thatf :X, σ → Y, τisI-fuzzifying measurable.

Corollary 4.4. LetX, σandY, τbe two measurable spaces, andf :XY is a mapping. Then

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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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