A note on mean value and dispersion of intuitionistic fuzzy events

: In this paper, we compare two deﬁnitions of mean value and dispersion for intuitionistic fuzzy events. We show the connection between these two deﬁnitions and we introduce some types of mean values induced by intuitionistic fuzzy state and by intuitionistic fuzzy probability.


Introduction
In connection with the studying of the limit theorems for the intuitionistic fuzzy observables there was a need to define the notion of mean value and dispersion. First, the notion of integrable intuitionistic fuzzy observable appeared in the paper [8]. There, the authors used the probability measures as a composition an intuitionistic fuzzy observable and intuitionistic fuzzy probability for definition. Later, B. Riečan defined an intuitionistic mean value as an integral with help of an intuitionistic fuzzy distribution function, see [14]. In this paper, we define the intuitionistic fuzzy mean value and the intuitionistic fuzzy dispersion with the help of intuitionistic fuzzy state and we show the connection between these two definitions. We study the intuitionistic fuzzy mean value in connection with the intuitionistic fuzzy probability, too.
We note that in the whole text we use a notation IF as an abbreviation for intuitionistic fuzzy.

Preliminary notions
In this section, we present the basic notions like an intuitionistic fuzzy event, an intuitionistic fuzzy state, an intuitionistic fuzzy observable and an intuitionistic fuzzy probability. Recall that the notion of intuitionistic fuzzy sets was introduced by K. T. Atanassov in 1983 as a generalization of Zadeh's fuzzy sets (see [1,2,3]).
The family of all IF -events on (Ω, S) will be denoted by F. The function µ A : Ω −→ [0, 1] will be called the membership function and the function ν A : Ω −→ [0, 1] will be called the non-membership function.
If A = (µ A , ν A ) ∈ F, B = (µ B , ν B ) ∈ F, then we define the Lukasiewicz binary operations ⊕, on F by and the partial ordering is given by The second basic notion in the probability theory is the notion of a state (see [14]).
The third basic notion in the probability theory is the notion of an observable. Let J be the family of all intervals in R of the form Then the σ-algebra σ(J ) is denoted B(R) and it is called the σ-algebra of Borel sets. Its elements are called Borel sets.
Definition 2.4. By an IF-observable on F we understand each mapping x : B(R) → F satisfying the following conditions: Similarly as in the classical case the following theorem can be proved ( [8,14]).
Sometimes we need to work with an intuitionistic fuzzy probability, see [9,10].
Definition 2.6. Let F be the family of all IF-events in Ω. A mapping P : F → J is called an IF-probability, if the following conditions hold: Of course, each P(A) is an interval, denote it by P(A) = [P (A), P (A)]. By this way we obtain two real functions and some properties of P can be characterized by some properties of P , P , see [11].

Mean value using an intuitionistic fuzzy state
In this section we show two definitions of an intuitionistic fuzzy mean value and compare them.
In the first case we are inspired by Kolmogorov case. If ξ : Ω → R is a random variable, then where P ξ (B) = P (ξ −1 (B)).
Definition 3.1. We say that an IF-observable x is an integrable IF-observable, if the integral R t dm x (t) exists. In this case we define IF-mean value If the integral R t 2 dm x (t) exists, then we define IF-dispersion D 2 (x) by the formula In Similarly as in the classical case the following theorem can be proved ( [14]). Proof. In [14] Proposition 3.2.
The following theorem says about a connection between the definition of IF -mean value and IF -dispersion in Definition 3.1 and the notion of IF -distribution function.
Proof. Since F is the IF -distribution function of the probability distribution m x , we have Similarly the other equality can be obtained.
Therefore, we can define IF -mean value and IF -dispersion of an IF -observable in another way, see [4].
Moreover, if there exists R t 2 dF(t), then we define the IF-dispersion D 2 (x) by the formula In the classical case, we sometimes work with random variable, which is the composition of a measurable function and a random variable. In this case, the mean value is defined as follows where F is the distribution function of a random variable ξ and g is a measurable function. In [14] B. Riečan defined intuitionistic fuzzy mean value for this case as follows

Mean value using an intuitionistic fuzzy probability
In this section, we explain the definition of an IF -mean value with the help of an IF -probability. Since the IF -probability can be decomposed on two IF -states (see Theorem 2.7), we can use the results from the previous section.
First, the notion of integrable observable in the sense of IF -probability appeared in [8]. There P x = P • x and P x = P • x.
Later, in [6] was defined the notion of a square integrable IF -observable and the notion of IF -dispersion and IF -mean value in the sense of IF -probability. We say that IF-observable x is integrable, if the integrals R t dP x (t), R t dP x (t) exist. Then the IF-mean values are defined by We say that IF-observable x is square integrable, if the integrals R t 2 dP x (t), R t 2 dP x (t) exist. Then the IF-dispersions are defined by Similarly, we can define an IF -distribution function induced by an IF -probability. There instead of an IF -state m is used IF -probability P, see [5].
for each t ∈ R is called IF-distribution function, where F , F : R → [0, 1] are the corresponding distribution functions.
Then F , F are distribution functions, and Proof. It follows from Definition 4.2 and Theorem 3.3.

Conclusion
The paper concerns the probability theory on intuitionistic fuzzy events. We studied an intuitionistic fuzzy mean value and an intuitionistic fuzzy dispersion in connection with the intuitionistic fuzzy state and the intuitionistic fuzzy probability. We showed the relation between two ways of defining the intuitionistic fuzzy mean value and the intuitionistic fuzzy dispersion for the intuitionistic fuzzy observable.