(𝐿,𝑀)-Fuzzy 𝜎-Algebras

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.

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∶𝐿^𝑋→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. 𝐿^𝑋 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 𝜒_𝐴. 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∈𝐿^𝑋 and 𝑎∈𝐿, we use the following notation:

𝐴_[𝑎]={𝑥∈𝑋∶𝐴(𝑥)⩾𝑎}.(1.1)

If 𝐿 is completely distributive, then we can define

𝐴^[𝑎]={𝑥∈𝑋∶𝑎∈_¨¨𝛼(𝐴(𝑥))}.(1.2)

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

Theorem 1.1 (see [4]). Let 𝑀 be a completely distributive lattice and {𝑎_𝑖∶𝑖∈Ω}⊆𝑀. Then (1)𝛼(⋀_𝑖∈Ω𝑎_𝑖)=⋃_𝑖∈Ω𝛼(𝑎_𝑖), that is, 𝛼 is an ⋀−⋃ map;(2)𝛽(⋁_𝑖∈Ω𝑎_𝑖)=⋃_𝑖∈Ω𝛽(𝑎_𝑖), 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)=𝑎0,,𝑥𝑥_¨¨∈𝐷;∈𝐷.(𝑎∨𝐷)(𝑥)=1𝑎,,𝑥𝑥∈_¨¨∈𝐷𝐷.;(1.3) Then for each 𝐿-fuzzy set 𝐴 in 𝐿^𝑋, it follows that 𝐴=_𝑎∈𝐿𝑎∧𝐴_[𝑎].(1.4)

Theorem 1.2 (see [5, 7, 10]). If 𝐿 is completely distributive, then for each 𝐿-fuzzy set 𝐴 in 𝐿^𝑋, 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 {𝐴_𝑖∶𝑖∈Ω} in 𝐿^𝑋, it is easy to see that _𝑖∈Ω𝐴_𝑖_[𝑎]=_𝑖∈Ω𝐴_𝑖_[𝑎].(1.5) If 𝐿 is completely distributive, then it follows [7] that _𝑖∈Ω𝐴_𝑖^[𝑎]=_𝑖∈Ω𝐴_𝑖^[𝑎].(1.6)

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:(1)for any constant fuzzy set 𝛼, 𝛼∈𝜎;(2)for any 𝐴∈[0,1]^𝑋, 1−𝐴∈𝜎;(3)for any {𝐴_𝑛∶𝑛∈ℕ}⊆𝜎, ⋁_𝑛∈ℕ𝐴_𝑛∈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 𝐿^𝑋 is called an 𝐿-𝜎-algebra if it satisfies the following three conditions:(1)for any 𝑎∈𝐿, constant 𝐿-fuzzy set 𝑎⋀𝜒_𝑋∈𝜎;(2)for any 𝐴∈𝐿^𝑋, 𝐴^∈𝜎;(3)for any {𝐴_𝑛∶𝑛∈ℕ}⊆𝜎, ⋁_𝑛∈ℕ𝐴_𝑛∈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 an 𝐿-𝜎-algebra 𝜎 is a crisp subset of 𝐿^𝑋. Now we consider an 𝑀-fuzzy subset 𝜎 of 𝐿^𝑋.

Definition. Let 𝑋 be a nonempty set. A mapping 𝜎∶𝐿^𝑋→𝑀 is called an (𝐿,𝑀)-fuzzy 𝜎-algebra if it satisfies the following three conditions: (LMS1)𝜎(𝜒_∅)=⊤_𝑀;(LMS2) for any 𝐴∈𝐿^𝑋, 𝜎(𝐴)=𝜎(𝐴^);(LMS3) for any {𝐴_𝑛∶𝑛∈ℕ}⊆𝐿^𝑋, 𝜎(⋁_𝑛∈ℕ𝐴_𝑛)≥⋀_𝑛∈ℕ𝜎(𝐴_𝑛).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⋀𝜒_𝑋)=⊤_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 𝐿^𝑋 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 {𝐴_𝑛∶𝑛∈ℕ}⊆𝜎, ⋁_𝑛∈ℕ𝐴_𝑛∈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, 𝜎(𝐴)=𝜎(𝐴^);(MS3) for any {𝐴_𝑛∶𝑛∈ℕ}⊆2^𝑋, 𝜎(⋁_𝑛∈ℕ𝐴_𝑛)≥⋀_𝑛∈ℕ𝜎(𝐴_𝑛).

Example 2.3. Let  (𝑋,𝜎) be a crisp measurable space. Define𝜒_𝜎∶𝟐^𝑋→[0,1] by 𝜒_𝜎(𝐴)=10,,𝐴𝐴_¨¨∈𝜎;∈𝜎.(2.1) Then it is easy to prove that (𝑋,𝜒_𝜎) is a [0,1]-fuzzifying measurable space.

Example 2.4. Let 𝑋 be a nonempty set and 𝜎∶𝟐^𝑋→[0,1] a mapping defined by 𝜎(𝐴)=10,.5,𝐴𝐴_¨¨∈{∅,𝑋};∈{∅,𝑋}.(2.2) Then it is easy to prove that (𝑋,𝜎) is a [0,1]-fuzzifying measurable space. If 𝐴∈𝟐^𝑋 with 𝐴∈_¨¨{∅,𝑋}, then 0.5 is the degree to which 𝐴 is measurable.

Example 2.5. Let 𝑋 be a nonempty set and 𝜎∶[0,1]^𝑋→[0,1] a mapping defined by 𝜎(𝐴)=1,𝐴∈𝜒^∅0.5,𝐴_¨¨,𝜒_𝑋;∈𝜒_∅,𝜒_𝑋.(2.3) 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_∅,𝜒_𝑋}, 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 {𝐴_𝑛∶𝑛∈ℕ}⊆𝐿^𝑋, 𝜎(⋀_𝑛∈ℕ𝐴_𝑛)≥⋀_𝑛∈ℕ𝜎(𝐴_𝑛).

Proof. This can be proved from the following fact: 𝜎_𝑛∈ℕ𝐴_𝑛=𝜎_𝑛∈ℕ𝐴_𝑛^≥_𝑛∈ℕ𝜎𝐴_𝑛^=_𝑛∈ℕ𝜎𝐴_𝑛.(2.4) The next two theorems give characterizations of an (𝐿,𝑀)-fuzzy 𝜎-algebra.

Theorem 2.7. A mapping 𝜎∶𝐿^𝑋→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∶𝟐^𝑋→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∶𝐿^𝑋→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∶𝐿^𝑋→𝑀 is an (𝐿,𝑀)-fuzzy 𝜎-algebra and 𝑎∈𝛼(⟂_𝑀). Now we prove that 𝜎^[𝑎] is an 𝐿-𝜎-algebra.(LS1) By 𝜎(𝜒_∅)=⊤_𝑀 and 𝛼(⊤_𝑀)=∅, we know that 𝑎∈_¨¨𝛼(𝜎(𝜒_∅)); this implies that 𝜒_∅∈𝜎^[𝑎].(LS2) If 𝐴∈𝜎^[𝑎], then 𝑎∈_¨¨𝛼(𝜎(𝐴))=𝛼(𝜎(𝐴^)); this shows that 𝐴^∈𝜎^[𝑎].(LS3) If {𝐴_𝑖∶𝑖∈Ω}⊆𝜎^[𝑎], then forall𝑖∈Ω, 𝑎∈_¨¨𝛼(𝜎(𝐴_𝑖)). Hence 𝑎∈_¨¨⋃_𝑖∈Ω𝛼(𝜎(𝐴_𝑖)). By 𝜎(⋁_𝑖∈Ω𝐴_𝑖)≥⋀_𝑖∈Ω𝜎(𝐴_𝑖), we know that𝛼𝜎_𝑖∈Ω𝐴_𝑖⊆𝛼_𝑖∈Ω𝜎𝐴_𝑖=_𝑖∈Ω𝛼𝜎𝐴_𝑖.(2.5) This shows that 𝑎∈_¨¨𝛼(𝜎(⋁_𝑖∈Ω𝐴_𝑖)). Therefore, ⋁_𝑖∈Ω𝐴_𝑖∈𝜎^[𝑎]. The proof is completed.

Corollary 2.10. If 𝑀 is completely distributive, then a mapping 𝜎∶𝟐^𝑋→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 (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. Let 𝑀 be completely distributive, and let {𝜎^𝑎∶𝑎∈𝛼(⟂_𝑀)} be a family of 𝐿-𝜎-algebras. If 𝜎^𝑎=⋂{𝜎^𝑏∶𝑎∈𝛼(𝑏)} for all 𝑎∈𝛼(⟂_𝑀), then there exists an (𝐿,𝑀)-fuzzy 𝜎-algebra 𝜎 such that 𝜎^[𝑎]=𝜎^𝑎.

Proof. Suppose that 𝜎^𝑎=⋂{𝜎^𝑏∶𝑎∈𝛼(𝑏)} for all 𝑎∈𝛼(⟂_𝑀). Define 𝜎∶𝐿^𝑋→𝑀 by 𝜎(𝐴)=_𝑎∈𝑀(𝑎∨𝜎^𝑎(𝐴))={𝑎∈𝑀∶𝐴∈_¨¨𝜎^𝑎}.(2.6) By Theorem 1.2, we can obtain that 𝜎^[𝑎]=𝜎^𝑎.

Corollary 2.13. Let 𝑀 be completely distributive, and let {𝜎^𝑎∶𝑎∈𝛼(⟂_𝑀)} be a family of 𝜎-algebras. If 𝜎^𝑎=⋂{𝜎^𝑏∶𝑎∈𝛼(𝑏)} for all 𝑎∈𝛼(⟂_𝑀), then there exists an 𝑀-fuzzifying 𝜎-algebra 𝜎 such that 𝜎^[𝑎]=𝜎^𝑎.

Theorem 2.14. Let 𝑀 be completely distributive, and let {𝜎_𝑎∶𝑎∈𝑀⧵{⟂_𝑀}} be a family of 𝐿-𝜎-algebra. If 𝜎_𝑎=⋂{𝜎_𝑏∶𝑏∈𝛽(𝑎)} for all 𝑎∈𝑀⧵{⟂_𝑀}, then there exists an (𝐿,𝑀)-fuzzy 𝜎-algebra 𝜎 such that 𝜎_[𝑎]=𝜎_𝑎.

Proof. Suppose that 𝜎_𝑎=⋂{𝜎_𝑏∶𝑏∈𝛽(𝑎)} for all 𝑎∈𝑀⧵{⟂_𝑀}. Define 𝜎∶𝐿^𝑋→𝑀 by 𝜎(𝐴)=_𝑎∈𝑀𝑎∧𝜎_𝑎(𝐴)=𝑎∈𝑀∶𝐴∈𝜎_𝑎.(2.7) By Theorem 1.2, we can obtain 𝜎_[𝑎]=𝜎_𝑎.

Corollary 2.15. Let 𝑀 be completely distributive, and let {𝜎_𝑎∶𝑎∈𝑀⧵{⟂_𝑀}} be a family of 𝜎-algebra. If 𝜎_𝑎=⋂{𝜎_𝑏∶𝑏∈𝛽(𝑎)} for all 𝑎∈𝑀⧵{⟂_𝑀}, then there exists an 𝑀-fuzzifying 𝜎-algebra 𝜎 such that 𝜎_[𝑎]=𝜎_𝑎.

Theorem 2.16. Let {𝜎_𝑖∶𝑖∈Ω} be a family of (𝐿,𝑀)-fuzzy 𝜎-algebra on 𝑋. Then _𝑖⋀_∈Ω𝜎_𝑖 is an (𝐿,𝑀)-fuzzy 𝜎-algebra on 𝑋, where ⋀_𝑖∈Ω𝜎_𝑖∶𝐿^𝑋→𝑀 is defined by (⋀_𝑖∈Ω𝜎_𝑖)(𝐴)=⋀_𝑖∈Ω𝜎_𝑖(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 𝑓^←𝐿(𝜏)∶𝐿^𝑋→𝑀 by forall𝐴∈𝐿^𝑋, 𝑓^←𝐿(𝜏)(𝐴)=𝜏(𝐵)∶𝑓^←𝐿(𝐵)=𝐴,where∀𝑥∈𝑋,𝑓^←𝐿(𝐵)(𝑥)=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_∅=𝜏(𝐵)∶𝑓^←𝐿(𝐵)=𝜒_∅=𝜏𝜒_∅=⊤_𝑀.(3.2)
(LMS2) can be shown from the following fact: forall𝐴∈𝐿^𝑋, 𝑓^←𝐿(𝜏)(𝐴)==𝜏𝜏(𝐵𝐵)∶𝑓^←𝐿(𝐵)=𝐴_=𝑓_←𝐿∶𝑓^←𝐿𝐵^=𝑓^←𝐿(𝐵)^=𝐴^(𝜏)𝐴^.(3.3)
(LMS3) for any {𝐴_𝑛∶𝑛∈ℕ}⊆𝐿^𝑋, by𝑓^←𝐿(𝜏)_𝑛∈ℕ𝐴_𝑛=𝜏(𝐵)∶𝑓^←𝐿(𝐵)=_𝑛∈ℕ𝐴_𝑛≥𝜏_𝑛∈ℕ𝐵_𝑛∶𝑓^←𝐿𝐵_𝑛=𝐴_𝑛≥_𝑛∈ℕ𝑓^←𝐿(𝜏)𝐴_𝑛(3.4)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(𝑓^←𝐿(𝐵))≥𝜏(𝐵) for all 𝐵∈𝐿^𝑌.
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 𝑓^←𝐿(𝜏)(𝐴)≤𝜎(𝐴) for all 𝐴∈𝐿^𝑋.

Proof. Necessity.. If 𝑓∶𝑋→𝑌 is (𝐿,𝑀)-fuzzy measurable, then 𝜎(𝑓^←𝐿(𝐵))≥𝜏(𝐵) for all 𝐵∈𝐿^𝑌. Hence for all 𝐵∈𝐿^𝑌, we have𝑓^←𝐿(𝜏)(𝐴)≤=𝜎𝜏(𝐵𝑓)∶𝑓^←𝐿(𝐵)=𝐴_←𝐿=𝜎(𝐴).(𝐵)∶𝑓^←𝐿(𝐵)=𝐴(3.5)Sufficiency.. If 𝑓^←𝐿(𝜏)(𝐴)≤𝜎(𝐴) for all 𝐴∈𝐿^𝑋, then 𝜏(𝐵)≤𝑓^←𝐿(𝜏)(𝑓^←𝐿(𝐵))≤𝜎(𝑓^←𝐿(𝐵)) for all 𝐵∈𝐿^𝑌; 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).

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)∶𝐼^𝑋→𝐼 by 𝜁(𝜎)(𝐴)=_𝐵∈ℬ𝜎𝐴⁻¹(𝐵).(4.1) 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⋀𝜒_𝑋)⁻¹(𝐵)=𝑋; if 𝑎∈_¨¨𝐵, then (𝑎⋀𝜒_𝑋)⁻¹(𝐵)=∅. However, we have that 𝜎((𝑎⋀𝜒_𝑋)⁻¹(𝐵))=1. This shows that 𝜁(𝜎)(𝑎⋀𝜒_𝑋)=1.
(LMS2) forall𝐴∈𝐼^𝑋 and forall𝐵∈ℬ, we have 𝜁(𝜎)𝐴^==_𝐵∈ℬ𝜎(1−𝐴)⁻¹(𝐵)_𝐵∈ℬ=𝜎({𝑥∈𝑋∶1−𝐴(𝑥)∈𝐵})_𝐵∈ℬ=𝜎({𝑥∈𝑋∶∃𝑏∈𝐵,s.t.𝐴(𝑥)=1−𝑏})_𝐵∈ℬ=𝜁(𝜎)𝜎(𝐴𝐴).⁻¹(𝐵)(4.2)
(LMS3) for any {𝐴_𝑛∶𝑛∈ℕ}⊆𝐿^𝑋 and forall𝐵∈ℬ, by 𝜁(𝜎)_𝑛∈ℕ𝐴_𝑛=_𝐵∈ℬ𝜎⎛⎜⎜⎝_𝑛∈ℕ𝐴_𝑛⁻¹=(𝐵)_𝐵∈ℬ⎞⎟⎟⎠≥𝜎_𝑛∈ℕ𝐴^𝑛−1(𝐵)_𝐵∈ℬ_𝑛∈ℕ=𝜎𝐴^𝑛−1(𝐵)_𝑛∈ℕ_𝐵_∈ℬ𝜎𝐴^𝑛−1(𝐵)=_𝑛∈ℕ𝜁(𝜎)𝐴_𝑛,(4.3)
we obtain 𝜁(𝜎)(⋁_𝑛∈ℕ𝐴_𝑛)≥⋀_𝑛∈ℕ𝜁(𝜎)(𝐴_𝑛).

Corollary 4.2. Let (𝑋,𝜎) be a measurable space. Define a subset 𝜁(𝜎)⊆𝐼^𝑋 (can be viewed as a mapping 𝜁(𝜎)∶𝐼^𝑋→𝟐) by 𝜁(𝜎)=𝐴∈𝐼^𝑋∶∀𝐵∈ℬ,𝐴⁻¹(𝐵)∈𝜎.(4.4) 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∈𝟐^𝑋. In order to prove that 𝑓∶(𝑋,𝜁(𝜎))→(𝑌,𝜁(𝜏)) is (𝐼,𝐼)-fuzzy measurable, we need to prove that 𝜁(𝜎)(𝑓^←𝐿(𝐴))≥𝜁(𝜏)(𝐴) for any 𝐴∈𝐼^𝑋.
In fact, for any 𝐴∈𝐼^𝑋, by 𝜁(𝜎)𝑓^←𝐿(𝐴)==_𝐵∈ℬ𝜎𝑓^←𝐿(𝐴)⁻¹(𝐵)=_𝐵∈ℬ𝜎(𝐴∘𝑓)⁻¹(𝐵)_𝐵∈ℬ≥𝜎(𝐵∘𝐴∘𝑓)=_𝐵∈ℬ𝜎𝑓⁻¹𝐴⁻¹(𝐵)_𝐵∈ℬ𝜏𝐴⁻¹(𝐵)=𝜁(𝜏)(𝐴),(4.5) we can prove the necessity.
Sufficiency. Suppose that 𝑓∶(𝑋,𝜁(𝜎))→(𝑌,𝜁(𝜏)) is (𝐼,𝐼)-fuzzy measurable. Then 𝜁(𝜎)(𝑓^←𝐼(𝐴))≥𝜁(𝜏)(𝐴) for any 𝐴∈𝐼^𝑋. In particular, it follows that 𝜁(𝜎)(𝑓^←𝐼(𝐴))≥𝜁(𝜏)(𝐴) for any 𝐴∈𝟐^𝑋. 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∈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⁻¹(𝐵)=𝐴 or 𝐴⁻¹(𝐵)=𝐴′. However, we have𝜎𝑓^←𝐼(𝐴)==𝜎𝜎𝑓𝑓^←𝐼_←𝐼(𝐴)=_𝐵∈ℬ(𝐴)∧𝜎𝑓^←𝐼(𝐴)^=𝜁(𝜎)𝜎𝑓_←𝐿𝑓^←𝐿(𝐴)⁻¹(𝐵)≥=𝜁𝜁((𝜏𝜏))((𝐴𝐴))∧(𝐴𝜁)(𝜏)𝐴_=_𝐵∈ℬ𝜏𝐴⁻¹(𝐵)=𝜏(𝐴).(4.6)
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).