m-almost everywhere convergence of intuitionistic fuzzy observables induced by Borel measurable function

Abstract: In paper [4] we studied the upper and the lower limits of sequence of intuitionistic fuzzy observables. We used an intuitionistic fuzzy state m for a definition the notion of almost everywhere convergence. We compared two concepts of m-almost everywhere convergence. The aim of this paper is to show the connection between almost everywhere convergence in classical probability space induced by Kolmogorov construction and m-almost everywhere convergence in intuitionistic fuzzy space. We studied the sequence of intuitionistic fuzzy observables induced by Borel measurable function.


Introduction
In [1][2][3] K. T. Atanassov introduced the notion of intuitionistic fuzzy sets. Then in [7] B. Riečan defined the intuitionistic fuzzy state on the family of intuitionistic fuzzy events where µ A , ν A are S-measurable functions, µ A , ν A : Ω → [0, 1], as a mapping m from the family F to the set R by the formula where P : S → [0, 1] is a probability measure and α ∈ [0, 1].
In paper [4] we defined the upper and the lower limits for sequence of intuitionistic fuzzy observables. We used an intuitionistic fuzzy state m for a definition the notion of almost everywhere convergence. We compared two concepts of m-almost everywhere convergence.
In this paper we study the m-almost everywhere convergence of sequence of intuitionistic fuzzy observables g n (x 1 , . . . , x n ) : B(R) → F given by where h n : B(R n ) → F is the joint intuitionistic fuzzy observable of intuitionistic fuzzy observables x 1 , . . . , x n and g n : R n → R is a Borel measurable function. We show the connection between m-almost everywhere convergence of this sequence of intuitionistic fuzzy observables and P -almost everywhere convergence of random variables in classical probability space induced by Kolmogorov construction.
Remark. Note that in a whole text we use a notation "IF" in short as the phrase "intuitionistic fuzzy."

IF-events, IF-states and IF-observables
First we start with definitions of basic notions.
and the partial ordering is then given by In paper we use max-min connectives defined by where a * = 1 − a.
If f = χ A , then the corresponding IF-set has the form In this case A ⊕ B corresponds to the union of sets, A B to the intersection of sets and ≤ to the set inclusion.
In the IF-probability theory [7,9] instead of the notion of probability, we use the notion of state.
Probably the most useful result in the IF-state theory is the following representation theorem ( [7]): 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 2.6. By an IF-observable on F we understand each mapping x : B(R) → F satisfying the following conditions: Remark 2.7. Sometimes we need to work with n-dimensional IF-observable x : B(R n ) → F defined as a mapping with the following conditions: If n = 1 we simply say that x is an IF-observable.
Similarly to the classical case, the following theorem can be proved ( [9]). Then m x : B(R) −→ [0, 1] is a probability measure.

Product operation, joint IF-observable and function of several IF-observables
In [5] we introduced the notion of product operation on the family of IF-events F and showed an example of this operation.
Definition 3.1. We say that a binary operation · on F is product if it satisfying the following conditions: (ii) the operation · is commutative and associative; The following theorem provides an example of product operation for IF-events.
In [8] B. Riečan defined the notion of a joint IF-observable and proved its existence.  If we have several IF-observables and a Borel measurable function, we can define the IFobservable, which is the function of several IF-observables. Regarding this we provide the following definition. for each A ∈ B(R).

Lower and upper limits, m-almost everywhere convergence
In [4] we defined the notions of lower and upper limits for a sequence of IF-observables and showed the connection between two kinds of m-almost everywhere convergence.
Note that if another IF-observable y satisfies the above condition, then m • y = m • x.
for every t ∈ R.
In accordance to Proposition 4.2 we can extend the notion of m-almost everywhere convergence in the following way.

Definition 4.4. A sequence (x n ) n of an IF-observables converges m-almost everywhere to an
for every t ∈ R.
5 P -almost everywhere convergence and m-almost everywhere convergence The main result of this section is given in Theorem 5.1. The main step is presented in the following proposition.
Recall, that the corresponding probability space is (R N , σ(C), P ), where C is the family of all sets of the form . . , t n ∈ A n }, and P is the probability measure determined by the equality The corresponding projections ξ n : R N → R are defined by the equality Proposition 5.1. Let (x n ) n be a sequence of IF-observables, (ξ n ) n the sequence of corresponding projections, g n : R n → R be a Borel measurable functions (n = 1, 2, . . .). Then Proof. We have The second inequality can be proved similarly.
Theorem 5.1. Let (x n ) n be a sequence of IF-observables, (ξ n ) n be the sequence of corresponding projections, (g n ) n be a sequence of Borel measurable functions g n : R n → R. If the sequence g n (ξ 1 , . . . , ξ n ) n converges P -almost everywhere, then the sequence g n (x 1 , . . . , x n ) n converges m-almost everywhere and for each t ∈ R.

Conclusion
The Theorem 5.1 is important for the proof of the Individual ergodic theorem in intuitionistic fuzzy case, where we work with the sequence of several IF-observables induced by the Borel function.