Basic Fuzzy Event Space and Probability Distribution of Probability Fuzzy Space

: In this paper, the problems of basic fuzzy event space and of probability fuzzy space are studied. Firstly, the concepts of basic fuzzy event, fuzzy event and basic fuzzy event space are deﬁned, related properties are investigated, and some results that will be used in the next study of probability fuzzy space are obtained. Then, the deﬁnitions of the probability function for fuzzy events and probability fuzzy space are given, some properties of the deﬁned probability function are obtained. In addition, some models of probability distribution of probability fuzzy space based on a known probability space are proposed, and some examples are given to show the usability of the proposed models of probability distribution.


Introduction
It is known that the probability and randomness theory is the theoretical basis for studying and dealing with the problem of the uncertainty of whether or not an event happened.The fuzzy set theory is the theoretical basis for studying and dealing with the problem of the uncertainty of the boundary of an event concept (for example, see [1,2]).The theory from combination of random theory and fuzzy set theory can be used to study and deal with the problem with these two mixed uncertainty attributes.
On the combination of random theory and fuzzy set theory, many scholars have engaged in, or are engaging in the work in this area.For example, in [3], Zadeh defined the probability of a fuzzy set as the expectation of its membership function; in [4], Khalili studied the independence between fuzzy events; in [5], Smets introduced the concept of the conditional probability of a fuzzy event; in [6], Baldwin, Lawry and Martin discussed the problem of the conditional probability of fuzzy subsets of a continuous domain; in [7], Buoncristiani investigated probability on -fuzzy sets; in [8], Lee and Li determined an order of fuzzy numbers (about fuzzy numbers, we can see [9][10][11][12]) based on the concept of probability measure of fuzzy events due to Zadeh; in [13], Heilpern defined the expected value of fuzzy variables, and investigated its properties; in [14], Gil gave a discussion on treating fuzziness as a kind of randomness in studying statistical management of fuzzy elements in random experiments; in [15], Flaminio and Godo proposed a logic for reasoning about the probability of fuzzy events; in [16], Kato, Izuka et al. proposed a new fuzzy probability distribution function containing fuzzy numbers as its parameters; in [17], Xia provided a fuzzy probability system, which has a more original theoretical starting point, and appears to deal with such uncertainty as has subjectivity and fuzziness; in [18], Talašová and Pavlačka defined a fuzzy probability space that enables an adequate mathematical modeling of expertly set uncertain probabilities of states of the world; in [19], Biacino extended the definition of belief function to fuzzy events starting from a basic assignment of probability on some fuzzy focal events and using a suitable notion of inclusion for fuzzy subsets; in [20,21], Kßißc . and Leblebicio ǧlu studied the problems of fuzzy discrete event systems; in [22], Kahraman and Kaya made an investment analysis by using the concept of probability of a fuzzy event.
Recently, there has still been a lot of work on the theory and application of the combination of randomness and fuzziness.For example, in 2012, Liu and Dziong formalized the notion of codiagnosability for decentralized diagnosis of fuzzy discrete-event systems, in which the observability of fuzzy events is defined to be fuzzy instead of crisp in [23]; in 2014, Purba, Lu, Zhang and Pedrycz developed a fuzzy reliability algorithm to effectively generate basic event failure probabilities without reliance on quantitative historical failure data through qualitative data processing in [24]; in 2015, Purba and Tjahyani et al. proposed a fuzzy probability based fault tree analysis to propagate and quantify epistemic uncertainty raised in basic event reliability evaluations to complement conventional fault tree analysis which can only evaluate aleatory uncertainty in [25]; in [26], Zhao and Hu took fuzzy probability and interval-valued fuzzy probability into consideration; in 2016, Lower, Magott and Skorupski presented a new approach as previous research in analyzing Air Traffic Incidents has focused more on defining accident occurrence probabilities in [27]; in [28], Chutia and Datta proposed the fuzzy random variable valued Gumbel, Weibull and Gaussian functions, and discussed fundamental properties of these functions in the fuzzy probability space; in 2017, Coletti, Petturiti and Vantaggi introduced the concept of possibility of a fuzzy event, and provided a comparison with the probability of a fuzzy event in [29].
While many results have been obtained in the theory and application of the combination of random theory and fuzzy set theory, the work has not yet reached a perfect degree.In the aspect of the theoretical research results, the researchers generally study the combination of the two theory only from the point of view of mathematics (only from the mathematical theory itself), this leads to that the obtained theoretical results may look (from the point of view of mathematics) very beautiful, but they lack application background, and it is difficult to get real application in engineering or practical problems.In the aspect of the applied research work, researchers often only focus on an isolated specific problem, the used theory (of combination of random theory and fuzzy set theory) is still in the initial stage, lacking in depth, and the obtained results are also lacking in systematicness.
In order to establish the systematic theory of "random fuzzy sets" and "random fuzzy numbers" with strong usability, in this paper, we do some preliminary research work.From the introduction of basic fuzzy events, we give some concepts such as basic fuzzy event space, fuzzy events, probability distribution on basic fuzzy event space, probability fuzzy space and so on, investigate their related properties, and propose some specific models of probability distribution of probability fuzzy space based on a known probability space, which have a strong application background.Specific arrangements are as follows: In Section 2, we briefly review some basic notions and definitions which will be used in this paper; in Section 3, we define the concepts of basic fuzzy event, fuzzy event and basic fuzzy event space, investigate related properties, and obtain some results that will be used in the next section; in Section 4, we introduce the definitions of the probability function about fuzzy events and probability fuzzy space, obtain some properties of the defined probability function, propose some models of probability distribution of probability fuzzy space based on a known probability space, and give some examples to show the using of the proposed models of probability distribution.In Section 5, we make a summary of this paper.

Basic Definition and Notation
Let Ψ be nonempty set (in this paper, we denote the empty set by φ).We denote the collection of all subsets of Ψ by 2 Ψ .A mapping µ : Ψ → [0, 1] is called a fuzzy subset (in short, a fuzzy set) of Ψ.We denote the collection of all fuzzy sets of Ψ by F (Ψ).
If σ(Ω) is a finite intersection event set, and the Condition (3) then we call P a finitely additive probability distribution function on σ(Ω), and say (Ω, σ(Ω), P) is a finitely additive probability space.

Basic Fuzzy Event Space
In order to establish the relevant theory of probability fuzzy space, in this section, we are going to give the following concepts of fuzzy basic events and fuzzy basic event space, and investigate their related properties: Let Ω be a basic event space.For ω ∈ Ω and r ∈ [0, 1], we define fuzzy set (ω, r) : and call it a basic fuzzy event of Ω.We denote Ω = {(ω, r) : ω ∈ Ω, r ∈ (0, 1]}, and call Ω the basic fuzzy event space (generated by Ω).

Definition 3.
Let Ω be a basic event space.We define mapping E : 2 Ω → F (Ω) as and call E canonical mapping of fuzzy events (with respect to Ω).

Remark 2.
Let Ω be a basic event space.For a Ã ∈ 2 Ω (i.e., Ã ⊂ Ω, we see that Ã is a classical set (collection) whose elements are some basic fuzzy events.However, according to the canonical mapping E, we can regard Ã as a fuzzy set of Ω (i.e., an element in 2 Ω).
For two nonempty sets A, B, we denote Let Ω be a basic event space.We have ( The canonical mapping E is not one-to-one mapping (see the following Example 1).

Proposition 3.
Let Ω be a basic event space, Ã ∈ S(2 Ω).Then About the mapping E S : S(2 Ω) → F (Ω), we have the following result: Proposition 4. Let Ω be a basic event space.Then E S is an injective mapping (i.e., one-to-one mapping).

Probability Fuzzy Space
Let Ω be a basic event space.By the definitions of basic fuzzy event space Ω (generated by Ω), r-level set of fuzzy set and strong r-level set of fuzzy set and Remark 2, we can see that for any Ã ∈ 2 Ω, supp Ã ∈ 2 Ω , and It is known that for a probability space (Ω, σ(Ω), P) (where, Ω is a basic event space, σ(Ω) ⊂ 2 Ω is a event space, P : σ(Ω) → [0, 1] is a probability function), the event space σ(Ω) should keep the closeness of operations of union, intersection and difference.However, due to the complexity of the structure of 2 Ω, it is often difficult to make a subset (i.e., a collection of some fuzzy events) of 2 Ω keeping the closeness of operations of union, intersection and difference.Therefore, when we introduce a probability fuzzy space ( Ω, σ( Ω), P), we do not claim that σ( Ω) (⊂ 2 Ω) keeps the closeness of operations of union, intersection and difference; we only claim that σ( Ω) satisfies supp Ã, [ Ã] r and ( Ã) r ∈ σ(Ω) for any Ã ∈ σ( Ω) and r ∈ [0, 1].
Owing to the complexity of structures of Ω and 2 Ω , and the non-closeness of operations of union, intersection and difference of σ( Ω), when we introduce a probability fuzzy space ( Ω, σ( Ω), P), if we still let probability function P satisfy P( Ω) = 1 and 0 ≤ P( Ã) ≤ 1 for any Ã ∈ σ( Ω), and the additivity of P: we define probability space (Ω, σ(Ω), P), then we can not guarantee the rationality of probability function P (see the following Example 3).
From the above analysis, we know that in order to define a rational probability function P on σ( Ω), we have to change this Condition (that is: to make it reasonable.Considering the complexity of the structures of Ω and σ( Ω), we propose to replace the irrational additivity of P with the following rational conditions: (1) P( Ã) ≤ P( B) for any Ã, B ∈ σ( Ω) with E( Ã) ⊂ E( B); Let Ω be a basic event space, Ω be the basic fuzzy event space generated by Ω, and then we call ( Ω, σ( Ω), P) a probability fuzzy space.Remark 6. (i) From the Conditions 2 and 3 in Definition 6, we see that P( Ã) ≤ 1 for any Ã ∈ σ( Ω).
Proof.The proof of the theorem is similar to the proof of Theorem 1, and we omit it.
Proof.The proof of the proposition is similar to the proof of Proposition 11, so we omit it.

Conclusions
In this paper, we studied the problems of basic fuzzy event space and of probability fuzzy space, the obtained results provide the basis for the future systematic establishment of the theory of "random fuzzy sets" and "random fuzzy numbers" with strong usability in the future.Firstly, we proposed the concepts of basic fuzzy event (Definition 1), and from the concept, we defined basic fuzzy event space (Definition 1), fuzzy events (Definition 2).In order to establish a reasonable probability distribution of fuzzy events (i.e., give reasonable definition of probability fuzzy space), we also defined the concepts of canonical mapping of fuzzy events (Definition 3) and simple fuzzy events, and obtained some results (Propositions 1-8) that will be used in studying probability distribution of fuzzy events.Then, we introduced the definitions of the probability function about fuzzy events and probability fuzzy space (Definition 6) and obtained some properties (Proposition 8) of the defined probability function.Then, we gave the model P(r) ( Ã) = P(( Ã) r ) (Theorem 1), model P[r] ( Ã) = P([ Ã] r ) (Theorem 2) and model P( Ã) = ∑ ω∈supp Ã P(ω) E( Ã) (ω) (Definition 7 and Theorem 4) for probability distribution of probability fuzzy space based on a known probability space, obtained some properties (Properties 2-4) of these probability distribution of probability fuzzy space, and gave some examples (Examples 4 and 5) to show the usability of the proposed models of probability distribution.
Author Contributions: Conceptualization and methodology and supervision, G.W.; formal analysis and investigation and writing-original draft preparation, G.W. and Y.X.; validation and writing-review and editing, Y.X. and S.Q.; funding acquisition, G.W. and S.Q.