Individual ergodic theorem for intuitionistic fuzzy observables using IF-probability

The aim of this paper is to formulate the individual ergodic theorem for intuitionistic fuzzy observables using P-almost everywhere convergence, where P is an intuitionistic fuzzy probability. Since the intuitionistic fuzzy probability can be decomposed to two intuitionistic fuzzy states, we can use the results holding for intuitionistic fuzzy states.


Introduction
In [1,2], K. T. Atanassov introduced the notion of intuitionistic fuzzy sets. Later P. Grzegorzewski and E. Mrówka defined the probability on the family of intuitionistic fuzzy events N = {(µ A , ν A ) ; µ A , ν A are S-measurable and µ A + ν A ≤ 1 Ω } as a mapping P from the family N to the set of all compact intervals in R by the formula where (Ω, S, P ) is the probability space, see [7]. This intuitionistic fuzzy probability was axiomatically characterized by B. Riečan (see [10]).
In this paper, we formulate the Individual ergodic theorem for intuitionistic fuzzy observables, using P-almost everywhere convergence, where P is an intuitionistic fuzzy probability. Recall that the formulation of the individual ergodic theorem for intuitionistic fuzzy events with product first appeared in the paper [3]. There we used a separating intuitionistic fuzzy probability. Since the intuitionistic fuzzy probability P can be decomposed to two intuitionistic fuzzy states, we can use the results holding for intuitionistic fuzzy states, which were proved in [6].
Remark that in a whole text we use a notation IF as an abbreviation for intuitionistic fuzzy.

IF-events, IF-states, IF-observables and IF-mean value
In this section we explain the basic notions from IF-probability theory, see [1,2,13,14,15].
The family of all IF-events on (Ω, S) will be denoted by F, µ A : Ω −→ [0, 1] will be called the membership function, ν A : Ω −→ [0, 1] will be called the non-membership function.
and the partial ordering is given by In the 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.
Consider a probability space (Ω, S, P ). Then in [7] the IF-probability P(A) of an IF-event A = (µ A , ν A ) ∈ F has been defined as a compact interval by the equality Let J be the family of all compact intervals. Then the mapping P : F → J can be defined axiomatically similarly as in [10].  and some properties of P can be characterized by some properties of P , P , see [11]. Proof. See [11,Theorem 2.3] Recall that by an intuitionistic fuzzy state (IF-state) m we understand each mapping m : F → [0, 1] which satisfies the following conditions (see [12]): Now we introduce 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 by B(R) and it is called the σ-algebra of Borel sets, its elements are called Borel sets (see [16]).
Definition 2.6. By an IF-observable on F we understand each mapping x : B(R) → F satisfying the following conditions: Similarly, we can define the notion of n-dimensional IF-observable.
Definition 2.7. By an n-dimensional IF-observable on F we understand each mapping x : B(R n ) → F satisfying the following conditions: Similarly, as in the classical case the following theorem can be proved ( [9,15]). Then m x : B(R) −→ [0, 1] is a probability measure.
Since m x : B(R) → [0, 1] plays now an analogous role as P ξ : B(R) → [0, 1], we can define IF-expected value E(x) by the same formula (see [9]). Definition 2.9. 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 the IF-expected value If the integral R t 2 dm x (t) exists, then we define IF-dispersion D 2 (x) by the formula

Product operation, joint IF-observable and function of several IF-observables
In [8] 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 satisfies the following conditions: (ii) The operation · is commutative and associative; The following theorem defines the product operation for IF-events.
In [13] B. Riečan defined the notion of a joint IF-observable and he proved its existence.  Remark 3.5. The joint IF-observable of the IF-observables x, y from Definition 3.3 is a twodimensional IF-observable.
If we have several IF-observables and a Borel measurable function, we can define the IF-observable, which is the function of several IF-observables. About this says the following definition.
Definition 3.6. Let x 1 , . . . , x n : B(R) → F be IF-observables, h n be their joint IF-observable and g n : R n → R be a Borel measurable function. Then, we define the IF-observable g n (x 1 , . . . , x n ) : B(R) → F by the formula for each A ∈ B(R).

Lower and upper limits, P-almost everywhere convergence
In [4] we defined the notions of lower and upper limits for a sequence of IF-observables.
In paper [5] we showed the connection between two kinds of P-almost everywhere convergence.
for every t ∈ R.
In accordance to Proposition 4.1, we can extend the notion of P-almost everywhere convergence in the following way.
for every t ∈ R.
Sometimes we need to work with a sequence of IF-observables induced by a Borel measurable function.
Recall, that the corresponding probability spaces are (R N , σ(C), P ) and (R N , σ(C), P ), where C is the family of all sets of the form and P , P are the probability measures determined by the equalities . The corresponding projections ξ n : R N → R are defined by the equality Theorem 4.7. 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 and P -almost everywhere, then the sequence g n (x 1 , . . . , x n ) n converges P-almost everywhere and for each t ∈ R. Moreover P lim sup n→∞ g n (x 1 , . . . , x n ) (−∞, t) = P E , P E for each t ∈ R, where E = {u ∈ R N : lim sup n→∞ g n ξ 1 (u), . . . , ξ n (u) < t}.

Individual Ergodic Theorem
In paper [5] we proved the modification of the classical Individual Ergodic Theorem using malmost everywhere convergence. Since the intuitionistic fuzzy probability P can be decomposed to two intuitionistic fuzzy states m (see [11,14]), then we try to formulate the modification of the classical Individual Ergodic Theorem using P-almost everywhere convergence. Now, we recall the modification of the Individual Ergodic Theorem for the IF-state (see [6]). We defined the IF-mean value of an IF-observable and P-almost everywhere convergence in the previous sections. Now we must define a transformation preserving an intuitionistic probability P.
Definition 5.2. Let (F, ·) be a family of IF-events with product, P be an IF-probability. Then, a mapping τ : F → F is said to be a P-preserving transformation if the following conditions are satisfied: (iii) If A n A, A n , A ∈ F, n ∈ N , then τ (A n ) τ (A); (iv) P τ (A) · τ (B) = P(A · B) for each A, B ∈ F. Now we show the connection to the m-preserving transformation. Recall that by m-preserving transformation we understand each mapping τ : F → F if the following conditions are satisfied: See [6].
Theorem 5.3. Let (F, ·) be a family of IF-events with product, P be an IF-probability. The mapping τ : F → F is the P-preserving transformation if and only if the mapping τ is the P -preserving transformation and the P -preserving transformation, where P , P are the IF-states.
Proof. "⇒" Let P be an IF-probability. Then by Theorem 2.5 it can be decomposed to two IF-states P , P such that P(A) = [P (A), P (A)] for each A ∈ F. If the mapping τ : F → F is the P-preserving transformation, then by (iv) from Definition 5. Hence, for each A, B ∈ F. Therefore, τ is a P -preserving transformation and a P -preserving transformation.
"⇐" The opposite direction can be proved similarly. Proof. Let P be an IF-probability. By Theorem 2.5 it can be decomposed to two IF-states P , P , such that P(A) = [P (A), P (A)] for each A ∈ F. Let τ be the P-preserving transformation. Then from Theorem 5.3 we obtain that τ is the P -preserving transformation and the P -preserving transformation, where P , P are the IF-states. Hence by Theorem 5.1 there exists an integrable IF-observable x * such that (τ i • x) = x * , P-almost everywhere.

Conclusion
The paper is concerned in ergodic theory for family of intuitionistic fuzzy events. We proved the Individual ergodic theorem for intuitionistic fuzzy observables using P-almost everywhere convergence, where P is an intuitinistic fuzzy probability. The results are a generalization of results given in [3].