Upper and lower limits and m -almost everywhere convergence of intuitionistic fuzzy observables

: In paper [4] we deﬁned the upper and the lower limits for sequence of intuitionistic fuzzy observables with the help of intuitionistic fuzzy probability P and we compared two concepts of P -almost everywhere convergence. The aim of this paper is to deﬁne the lower and upper limits using the intuitionistic fuzzy state m . We study two concepts of m -almost everywhere convergence and we show that they are equivalent, too.


Introduction
In [1,2] K. T. Atanassov introduced the notion of intuitionistic fuzzy sets. Then in [3] Grzegorzewski and Mrówka defined the probability on the family of intuitionistic fuzzy events where µ A , ν A are S-measurable, 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. This IF-probability was axiomatically characterized by B. Riečan (see [5]).
In paper [4] we defined the upper and the lower limits for sequence of intuitionistic fuzzy observables with the help of intuitionistic fuzzy probability P and we compared two concepts of P-almost everywhere convergence, where m is an intuitionistic fuzzy state. Since the intuitionistic fuzzy probability P can be decomposed to two intuitionistic fuzzy states, it is usefull to study m-almost everywhere convergence. In this paper we define the lower and upper limits for a sequence of intuitionistic fuzzy observables and we study two concepts of m-almost everywhere convergence, too.
Remark that in a whole text we use a notation "IF" for short a phrase "intuitionistic fuzzy".

IF-events, IF-states and IF-observables
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] be called the non-membership function.
If A = (µ A , ν A ) ∈ F, B = (µ B , ν B ) ∈ F, then we define the Łukasiewicz binary operations ⊕, on F by and the partial ordering is given by In the paper we use the 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 product of sets, and ≤ to the set inclusion.
In the IF -probability theory ( [7], [8]) 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 ( [6]): 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: Similarly as in the classical case the following theorem can be proved ( [8]). Then m x : B(R) −→ [0, 1] is a probability measure.

Almost everywhere convergence
Recall that a sequence (ξ n ) n of random variables converges almost everywhere to 0, if Similarly we can work with a sequence of IF -observables on (F, m).
Remark 4.4. The zero IF-observable 0 F can be rewritten in the following form for each A = (−∞, t) and t ∈ R.
for every t ∈ R.

Conclusion
The paper is concerned in a modification of m-almost everywhere convergence with the help of lim sup and lim inf and the construction of translations formulas for these limits in an intuitionistic fuzzy case. We compare two kinds of definition of the m-almost everywhere convergence for intuitionistic fuzzy events and we show that they are equivalent.