Conditional intuitionistic fuzzy probability and martingale convergence theorem using IF-probability

: The aim of this paper is to formulate the conditional intuitionistic fuzzy probability and a version of martingale convergence theorem with respect 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 probability space, see [5]. This intuitionistic fuzzy probability was axiomatically characterized by B. Riečan (see [8]).
Later in [6] K. Lendelová defined the conditional intuitionistic fuzzy probability only for separating intuitionistic fuzzy probability and she showed their properties in this case.
In [13] B. Riečan introduced the conditional intuitionistic fuzzy probability p(A | x) as a Borel measurable function f (i.e. B ∈ B(R) =⇒ f −1 (B) ∈ B(R)) such that for each B ∈ B(R). There A ∈ F, x : B(R) → F is intuitionistic fuzzy observable and m is the intuitionistic fuzzy state defined by B. Riečan 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] (see [10]).
In paper [4] we studied properties of the conditional intuitionistic fuzzy probability with respect to the intuitionistic fuzzy state and we formulated martingale convergence in this case. Since the intuitionistic fuzzy probability can be decomposed to two intuitionistic fuzzy states, we can use the results holding for intuitionistic fuzzy states. Therefore, we try to formulate the conditional intuitionistic fuzzy probability and a version of martingale convergence theorem with respect an intuitionistic fuzzy probability.
Remark that in a whole text we use a notation IF as an abbreviation for intuitionistic fuzzy.
Definition 2.2. Start with a measurable space (Ω, S). Hence S is a σ-algebra of subsets of Ω.
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 Consider a probability space (Ω, S, P ). Then in [5] 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 [8].  and some properties of P can be characterized by some properties of P , P , see [9]. Recall that by an intuitionistic fuzzy state (IF-state) m we understand each mapping m : F → [0, 1] which satisfies the following conditions (see [10]): 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 [14]).
Definition 2.5. 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 2.6. 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 ( [7,13]). 3 Joint IF-observable, function of several IF-observables and P-almost everywhere convergence In [6] 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: (i) (1 Ω , 0 Ω ) · (a 1 , a 2 ) = (a 1 , a 2 ) for each (a 1 , a 2 ) ∈ F; (ii) The operation · is commutative and associative; In the following theorem is the example of product operation for IF-events.
Theorem 3.2. The operation · defined by for each (x 1 , y 1 ), (x 2 , y 2 ) ∈ F is product operation on F.
In [11] B. Riečan defined the notion of a joint IF-observable and he proved its existence.   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 their joint IF-observable and g n : R n → R a Borel measurable function. Then we define the IF-observable g n (x 1 , . . . , x n ) : B(R) → F by the formula g n (x 1 , . . . , x n )(A) = h n g −1 n (A) . for each A ∈ B(R).
There exists connection between P-almost everywhere convergence and m-almost everywhere convergence (see [3]).
for each B ∈ B(R).
In paper [4] we proved the properties of the conditional IF-probability and the martingale convergence theorem in this case. (ii) 0 ≤ p A|y) ≤ 1 m y -almost everywhere; (iv) If A n A, then p A n |y p A|y m y -almost everywhere.

Conditional IF-probability and martingale convergence theorem using IF-probability
In this section we formulate modification of the conditional probability and the martingale convergence theorem with respect to the IF-probability in intuitionistic fuzzy case. p (A | y) dP y = P A · y(B) = P A · y(B) , P A · y(B) .
for each B ∈ B(R). Therefore by Definition 4.1 p (A | y), p (A | y) : R → R are the conditional IF-probabilities with respect to the IF-states P , P respectively. "⇐" It can be proved in the opposite direction as in "⇒". (ii) 0 ≤ p A|y) ≤ 1 P y -almost everywhere; p (A i |y) P y -almost everywhere.
(iv) If A n A, then p A n |y p A|y P y -almost everywhere.
Proof. The proof is straightforward using Theorem 5.2, Theorem 4.2 and Theorem 3.7.
Theorem 5.4 (Martingale convergence theorem). Let F be a family of IF-events with product · , A ∈ F, y : B(R) → F be an IF-observable, P be an IF-probability, g, g n : R → R (n = 1, 2, . . .) be the Borel measurable functions such that g −1 n B(R) g −1 B(R) . Then Proof. By Definition 5.1 and Theorem 5.2 we have where p (A | y • g −1 n ), p (A | y • g −1 n ) : R → R and p (A | y • g −1 ), p (A | y • g −1 ) : R → R are the conditional IF-probabilities with respect to the IF-states P , P respectively.

Conclusion
The paper is concerned in the probability theory on intuitionistic fuzzy sets. We defined the conditional intuitionistic fuzzy probability induced by an intuitionistic fuzzy probability and we proved its properties. We showed the connection between conditioning with respect to the intuitionistic fuzzy probability and conditioning with respect to the intuitionistic fuzzy state. We formulated and proved the martingale convergence theorem for the conditional intuitionistic fuzzy probability with respect to the intuitionistic fuzzy probability, too.