# (L,M)-Fuzzy σ-Algebras

Authors: Fu-Gui Shi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## 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.


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## Body

## 1. Introduction and Preliminaries

In 1980, Klement established an axiomatic theory of fuzzyσ-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 bothL and M are complete lattices, we define an (L,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.Whenσ 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, bothL 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,b∈M, a≺b if and only if for every subset D⊆M, the relation b⩽supD always implies the existence of d∈D with a⩽d [3]. {a∈M:a≺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)={a∈M:a≺opb}. In a completely distributive lattice M, there exist α(b) and β(b) for each b∈M, and 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)=∅ and α(1)=∅.For a complete latticeL, A∈LX and a∈L, we use the following notation:(1.1)A[a]={x∈X:A(x)⩾a}.IfL is completely distributive, then we can define(1.2)A[a]={x∈X:a∈α(A(x))}.Some properties of these cut sets can be found in [5–10].Theorem 1.1 (see [4]).
LetM be a completely distributive lattice and {ai:i∈Ω}⊆M. Then (1)
α(⋀i∈Ωai)=⋃i∈Ωα(ai), that is, α is an ⋀-⋃ map;(2)
β(⋁i∈Ωai)=⋃i∈Ωβ(ai), that is, β is a union-preserving map.For a∈L and D⊆X, we define two L-fuzzy sets a⋀D and a∨D as follows:
(1.3)(a⋀D)(x)={a,x∈D;0,x∈D.(a∨D)(x)={1,x∈D;a,x∈D.
Then for each L-fuzzy set A in LX, it follows that
(1.4)A=⋁a∈L(a⋀A[a]).Theorem 1.2 (see [5, 7, 10]).
IfL is completely distributive, then for each L-fuzzy set A in LX, we have  (1)
A=⋁a∈L(a⋀A[a])=⋀a∈L(a∨A[a]);(2)
foralla∈L, A[a]=⋂b∈β(a)A[b];(3)
foralla∈L, A[a]=⋂a∈α(b)A[b].For a family of L-fuzzy sets {Ai:i∈Ω} in LX, it is easy to see that
(1.5)(⋀i∈ΩAi)[a]=⋂i∈Ω(Ai)[a].
If L is completely distributive, then it follows [7] that
(1.6)(⋀i∈ΩAi)[a]=⋂i∈Ω(Ai)[a].Definition 1.3.
LetX 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 setα, α∈σ;(2)
for anyA∈[0,1]X, 1-A∈σ;(3)
for any{An:n∈ℕ}⊆σ, ⋁n∈ℕAn∈σ.The fuzzy sets in σ are called fuzzy measurable sets, and the pair (X,σ) a fuzzy measurable space.Definition 1.4.
LetL 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:(1)
for anya∈L, constant L-fuzzy set a⋀χX∈σ;(2)
for anyA∈LX, A′∈σ;(3)
for any{An:n∈ℕ}⊆σ, ⋁n∈ℕAn∈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 anL-σ-algebra σ is a crisp subset of LX. Now we consider an M-fuzzy subset σ of LX.Definition.
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)
σ(χ∅)=⊤M;(LMS2)
for anyA∈LX, σ(A)=σ(A′);(LMS3)
for any{An:n∈ℕ}⊆LX, σ(⋁n∈ℕAn)≥⋀n∈ℕσ(An).An (L,M)-fuzzy σ-algebra σ is said to be stratified if and only if it satisfies the following condition:
(LMS1)*   ∀a∈L, σ(a⋀χX)=⊤M.
Ifσ 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.
Ifσ 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 subsetσ of LX is regarded as a mapping σ:LX→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∈ℕAn∈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, whenL=2, a mapping σ:2X→M is an M-fuzzifying σ-algebra if and only if it satisfies the following conditions:
(MS1)
σ(∅)=⊤M;(MS2)
for anyA∈2X, σ(A)=σ(A′);(MS3)
for any{An:n∈ℕ}⊆2X, σ(⋁n∈ℕAn)≥⋀n∈ℕσ(An).Example 2.3.
Let(X,σ) be a crisp measurable space. Defineχσ:2X→[0,1] by
(2.1)χσ(A)={1,A∈σ;0,A∈σ.
Then it is easy to prove that (X,χσ) is a [0,1]-fuzzifying measurable space.Example 2.4.
LetX be a nonempty set and σ:2X→[0,1] a mapping defined by
(2.2)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 A∈2X 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→[0,1] a mapping defined by
(2.3)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∈ℕ}⊆LX, σ(⋀n∈ℕAn)≥⋀n∈ℕσ(An).Proof.
This can be proved from the following fact:(2.4)σ(⋀n∈ℕAn)=σ(⋁n∈ℕ(An)′)≥⋀n∈ℕσ((An)′)=⋀n∈ℕσ(An).
The next two theorems give characterizations of an (L,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}, σ[a] is an L-σ-algebra.Proof.
The proof is obvious and is omitted.Corollary 2.8.
A mappingσ:2X→M is an M-fuzzifying σ-algebra if and only if for each a∈M∖{⊥M}, σ[a] is a σ-algebra.Theorem 2.9.
IfM is completely distributive, then a mapping σ:LX→M is an (L,M)-fuzzy σ-algebra if and only if for each a∈α(⊥M), σ[a] 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 σ[a] is an L-σ-algebra.(LS1)
Byσ(χ∅)=⊤M and α(⊤M)=∅, we know that a∈α(σ(χ∅)); this implies that χ∅∈σ[a].(LS2)
IfA∈σ[a], then a∈α(σ(A))=α(σ(A′)); this shows that A′∈σ[a].(LS3)
If{Ai:i∈Ω}⊆σ[a], then foralli∈Ω, a∈α(σ(Ai)). Hence a∈⋃i∈Ωα(σ(Ai)). By σ(⋁i∈ΩAi)≥⋀i∈Ωσ(Ai), we know that(2.5)α(σ(⋁i∈ΩAi))⊆α(⋀i∈Ωσ(Ai))=⋃i∈Ωα(σ(Ai)).
This shows that a∈α(σ(⋁i∈ΩAi)). Therefore, ⋁i∈ΩAi∈σ[a]. The proof is completed.Corollary 2.10.
IfM is completely distributive, then a mapping σ:2X→M is an M-fuzzifying σ-algebra if and only if for each a∈α(⊥M), σ[a] is a σ-algebra.Now we consider the conditions that a family ofL-σ-algebras forms an (L,M)-fuzzy σ-algebra. By Theorem 1.2, we can obtain the following result.Corollary 2.11.
IfM is completely distributive, and σ is an (L,M)-fuzzy σ-algebra, then  (1)
σ[b]⊆σ[a] for any a,b∈M∖{⊥M} with a∈β(b);(2)
σ[b]⊆σ[a] for any a,b∈α(⊥M) with b∈α(a).Theorem 2.12.
LetM be completely distributive, and let {σa:a∈α(⊥M)} be a family of L-σ-algebras. If σa=⋂{σb:a∈α(b)} for all a∈α(⊥M), then there exists an (L,M)-fuzzy σ-algebra σ such that σ[a]=σa.Proof.
Suppose thatσa=⋂{σb:a∈α(b)} for all a∈α(⊥M). Define σ:LX→M by
(2.6)σ(A)=⋀a∈M(a∨σa(A))=⋀{a∈M:A∈σa}.
By Theorem 1.2, we can obtain that σ[a]=σa.Corollary 2.13.
LetM be completely distributive, and let {σa:a∈α(⊥M)} be a family of σ-algebras. If σa=⋂{σb:a∈α(b)} for all a∈α(⊥M), then there exists an M-fuzzifying σ-algebra σ such that σ[a]=σa.Theorem 2.14.
LetM be completely distributive, and let {σa:a∈M∖{⊥M}} be a family of L-σ-algebra. If σa=⋂{σb:b∈β(a)} for all a∈M∖{⊥M}, then there exists an (L,M)-fuzzy σ-algebra σ such that σ[a]=σa.Proof.
Suppose thatσa=⋂{σb:b∈β(a)} for all a∈M∖{⊥M}. Define σ:LX→M by
(2.7)σ(A)=⋁a∈M(a⋀σa(A))=⋁{a∈M:A∈σa}.
By Theorem 1.2, we can obtain σ[a]=σa.Corollary 2.15.
LetM be completely distributive, and let {σa:a∈M∖{⊥M}} be a family of σ-algebra. If σa=⋂{σb:b∈β(a)} for all a∈M∖{⊥M}, then there exists an M-fuzzifying σ-algebra σ such that σ[a]=σa.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)-fuzzy σ-algebra on X, where ⋀i∈Ωσi:LX→M is defined by (⋀i∈Ωσi)(A)=⋀i∈Ωσi(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(Y,τ) be an (L,M)-fuzzy measurable space and f:X→Y a mapping. Define a mapping fL←(τ):LX→M by forallA∈LX,
(3.1)fL←(τ)(A)=⋁{τ(B):fL←(B)=A},where∀x∈X,fL←(B)(x)=B(f(x)).
Then (X,fL←(τ)) is an (L,M)-fuzzy measurable space.Proof.
(LMS1) holds from the following equality:
(3.2)fL←(τ)(χ∅)=⋁{τ(B):fL←(B)=χ∅}=τ(χ∅)=⊤M.
(LMS2) can be shown from the following fact: forallA∈LX,
(3.3)fL←(τ)(A)=⋁{τ(B):fL←(B)=A}=⋁{τ(B′):fL←(B′)=fL←(B)′=A′}=fL←(τ)(A′).
(LMS3)
for any {An:n∈ℕ}⊆LX, by(3.4)fL←(τ)(⋁n∈ℕAn)=⋁{τ(B):fL←(B)=⋁n∈ℕAn}≥⋁{τ(⋁n∈ℕBn):fL←(Bn)=An}≥⋀n∈ℕfL←(τ)(An)wecanprove(LMS3).Definition 3.2.
Let(X,σ) and (Y,τ) be (L,M)-fuzzy measurable spaces. A mapping f:X→Y is called (L,M)-fuzzy measurable if σ(fL←(B))≥τ(B) for 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.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:X→Y is (L,M)-fuzzy measurable if and only if fL←(τ)(A)≤σ(A) for all A∈LX.Proof.
Necessity..
Iff:X→Y is (L,M)-fuzzy measurable, then σ(fL←(B))≥τ(B) for all B∈LY. Hence for all B∈LY, we have(3.5)fL←(τ)(A)=⋁{τ(B):fL←(B)=A}≤⋁{σ(fL←(B)):fL←(B)=A}=σ(A).Sufficiency..
IffL←(τ)(A)≤σ(A) for all A∈LX, then τ(B)≤fL←(τ)(fL←(B))≤σ(fL←(B)) for 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) and f:(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 if f:(X,σ[a])→(Y,τ[a]) is L-measurable for any a∈M∖{⊥M}.Theorem 3.6.
LetM 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 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) is M-fuzzifying measurable if and only if f:(X,σ[a])→(Y,τ[a]) is measurable for any a∈M∖{⊥M}.Corollary 3.8.
LetM 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 a∈α(⊥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):IX→I by
(4.1)ζ(σ)(A)=⋀B∈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∈ℬ and for any a∈I, if a∈B, then (a⋀χX)-1(B)=X; if a∈B, then (a⋀χX)-1(B)=∅. However, we have that σ((a⋀χX)-1(B))=1. This shows that ζ(σ)(a⋀χX)=1.
(LMS2)forallA∈IX and forallB∈ℬ, we have
(4.2)ζ(σ)(A′)=⋀B∈ℬσ((1-A)-1(B))=⋀B∈ℬσ({x∈X:1-A(x)∈B})=⋀B∈ℬσ({x∈X:∃b∈B,s.t.A(x)=1-b})=⋀B∈ℬσ(A-1(B))=ζ(σ)(A).
(LMS3) for any {An:n∈ℕ}⊆LX and forallB∈ℬ, by
(4.3)ζ(σ)(⋁n∈ℕAn)=⋀B∈ℬσ((⋁n∈ℕAn)-1(B))=⋀B∈ℬσ(⋃n∈ℕAn-1(B))≥⋀B∈ℬ⋀n∈ℕσ(An-1(B))=⋀n∈ℕ⋀B∈ℬσ(An-1(B))=⋀n∈ℕζ(σ)(An),
we obtainζ(σ)(⋁n∈ℕAn)≥⋀n∈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):IX→2) by
(4.4)ζ(σ)={A∈IX:∀B∈ℬ,A-1(B)∈σ}.
Then ζ(σ) is a stratified I-σ-algebra.From Corollary4.2, 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:X→Y 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 thatf:(X,σ)→(Y,τ) is I-fuzzifying measurable. Then σ(f-1(A))≥τ(A) for any A∈2X. 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 A∈IX.
In fact, for anyA∈IX, by
(4.5)ζ(σ)(fL←(A))=⋀B∈ℬσ((fL←(A))-1(B))=⋀B∈ℬσ((A∘f)-1(B))=⋀B∈ℬσ(B∘A∘f)=⋀B∈ℬσ(f-1(A-1(B)))≥⋀B∈ℬτ(A-1(B))=ζ(τ)(A),
we can prove the necessity.
Sufficiency.
Suppose thatf:(X,ζ(σ))→(Y,ζ(τ)) is (I,I)-fuzzy measurable. Then ζ(σ)(fI←(A))≥ζ(τ)(A) for any A∈IX. In particular, it follows that ζ(σ)(fI←(A))≥ζ(τ)(A) for any A∈2X. 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 A∈2X. In fact, for any A∈2X 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-1(B)=A or A-1(B)=A'. However, we have(4.6)σ(fI←(A))=σ(fI←(A))=σ(fI←(A))⋀σ(fI←(A)′)=⋀B∈ℬσ((fL←(A))-1(B))=ζ(σ)(fL←(A))≥ζ(τ)(A)=ζ(τ)(A)⋀ζ(τ)(A′)=⋀B∈ℬτ(A-1(B))=τ(A).
This shows thatf:(X,σ)→(Y,τ) is I-fuzzifying measurable.Corollary 4.4.
Let(X,σ) and (Y,τ) be two measurable spaces, and f:X→Y 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.



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