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

A[a]={x∈X:A(x)⩾a}.

If L is completely distributive, then we can define

A[a]={x∈X:a∈α(A(x))}.

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

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

Let M be a completely distributive lattice and {ai:i∈Ω}⊆M. Then

α(⋀i∈Ωai)=⋃i∈Ωα(ai), that is, α is an ⋀-⋃ map;

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:
(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
A=⋁a∈L(a⋀A[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=⋁a∈L(a⋀A[a])=⋀a∈L(a∨A[a]);

foralla∈L, A[a]=⋂b∈β(a)A[b];

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
(⋀i∈ΩAi)[a]=⋂i∈Ω(Ai)[a].
If L is completely distributive, then it follows [7] that
(⋀i∈ΩAi)[a]=⋂i∈Ω(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, α∈σ;

for any A∈[0,1]X, 1-A∈σ;

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.

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 a∈L, constant L-fuzzy set a⋀χX∈σ;

for any A∈LX, A′∈σ;

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. <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:LX→M is called an (L,M)-fuzzy σ-algebra if it satisfies the following three conditions:

σ(χ∅)=⊤M;

for any A∈LX, σ(A)=σ(A′);

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.

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:LX→2, then σ is an L-σ-algebra if and only if it satisfies the following conditions:

χ∅∈σ;

A∈σ⇒A′∈σ;

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, when L=2, a mapping σ:2X→M is an M-fuzzifying σ-algebra if and only if it satisfies the following conditions:

σ(∅)=⊤M;

for any A∈2X, σ(A)=σ(A′);

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
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.

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 A∈2X 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∈ℕ}⊆LX, σ(⋀n∈ℕAn)≥⋀n∈ℕσ(An).

Proof.

This can be proved from the following fact:
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 mapping σ: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 mapping σ: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.

If M 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 that σ:LX→M is an (L,M)-fuzzy σ-algebra and a∈α(⊥M). Now we prove that σ[a] is an L-σ-algebra.

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

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

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

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.

If M 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 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,b∈M∖{⊥M} with a∈β(b);

σ[b]⊆σ[a] for any a,b∈α(⊥M) with b∈α(a).

Theorem 2.12.

Let M 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 that σa=⋂{σb:a∈α(b)} for all a∈α(⊥M). Define σ:LX→M by
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.

Let M 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.

Let M 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 that σa=⋂{σb:b∈β(a)} for all a∈M∖{⊥M}. Define σ:LX→M by
σ(A)=⋁a∈M(a⋀σa(A))=⋁{a∈M:A∈σa}.
By Theorem 1.2, we can obtain σ[a]=σa.

Corollary 2.15.

Let M 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).

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,
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:
fL←(τ)(χ∅)=⋁{τ(B):fL←(B)=χ∅}=τ(χ∅)=⊤M.

(LMS2) can be shown from the following fact: forallA∈LX,
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, byfL←(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..

If f:X→Y is (L,M)-fuzzy measurable, then σ(fL←(B))≥τ(B) for all B∈LY. Hence for all B∈LY, we have

If fL←(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.

If f:(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.

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 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.

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 a∈α(⊥M).

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
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
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
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 obtain ζ(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
ζ(σ)={A∈IX:∀B∈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: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 that f:(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 any A∈IX, by
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 that f:(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

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: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.

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).

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