An Independence Property for General Information

The aim of this paper was a generalization of independence property proposed by J. Kampé de Feriét and B. Forte in Information Theory without probability, called general information. Therefore, its application to fuzzy sets has been presented.


Introduction
Since 1967-69, J. Kampé de Ferét and B. Forte have introduced, by axiomatic way, new information measures without probability [1]- [3]; later, in analogous way, with P. Benvenuti we have defined information measures without probability or fuzzy measure [4] for fuzzy sets [5] [6].This form of information measure is again called general information.
In Information Theory an important role has played by an independence property with respect to a given information measures J applied to crisp sets [7].These sets are called J-independent (i.e.independent each other with the respect to J) [8].
For this reason we will propose a generalization of J-independence property.The paper develops in the following way: in Section 2 we recall some preliminaires; in Section 3 the generalization of J-indepedence is proposed; the result is extended to fuzzy sets in Section 4. Section 5 is devoted to the conclusion.

Preliminaires
Let Ω be an abstract space and  the σ-algebra of crisp sets C ⊂ Ω , such that ( ) , Ω  is a measurable space.We refer to [7] for all knoledge and operations among crisp sets.
J. Kampé de Ferét and B. Forte gave the following definition [1] [2]: Definition 2.1 Measure of general information J for crisp sets is a mapping , C C satisfies the (iii), we say that 1 C and 2 C are J-independent, i.e. independent each other with respect to information J.

A Generalization of the J-Independence Property
In this paragraph we are going to present a generalization of the J-independence property.
We propose the following: Definition 3.1 Given a general information J, let 1 C and 2 C be two crisp sets in C such that 1 2 .

C C ∩ ≠ ∅
We say that 1 C and 2 C are J-idependent each other if there exists a continuous function We shall characterize the function Φ , taking into account the properties of the intersection for every ( ) .
the properties [(p 1 ) -(p 5 )] have translated in the following system of functional equations and inequalities [9] [10]: We can give the following Proposition 3.2 A class of solutions of the system [(P 1 ) - where h is any continuous, strictly increasing function


So, from (2), we have Proposition 3. 3 The generalization of the J-independence property for crisp sets is where h is any continuous, strictly increasing function


Remark When h is linear, the generalization (3) coincide the property (iii).

Extension to Fuzzy Setting
In this paragraph, we are considering the extension of J-independence property at fuzzy setting.
Let Ω be an abstract space and  the σ-algebra of fuzzy sets such that ( ) , Ω  is a measurable space [5], [6].In [4] we have given the definition of measure of general information for fuzzy sets: Definition 4.1 Measure of general information in fuzzy setting is a mapping , F F satisfies the (iii'), we say that 1 F and 2 F are J'-independent, i.e. independent each other with respect to information J ′ .Also in fuzzy setting, we generalize the (iii'), setting The properties of the intersection between fuzzy sets are the similar to the [(p 1 ) − (p 4 )] [5] [6].Therefore, we are looking for functions (4) solutions of the system [(P 1 ) − (P 5 )].We have again the similar result: Proposition 4.2 A class of solution of the system [(P 1 ) − (P 5 )] is where k is any continuous, strictly increasing function where k is any continuous, strictly increasing function The proof is similar to that given for crisp sets.


Remark.When k is linear, the generalization (6) coincide with the property (iii').

Conclusions
In this paper we have proposed a genralization of J-independence property between crisp sets: ( ) where h is any continuous, strictly increasing function ( ) where k is any continuous, strictly increasing function

=
and ( ). k +∞ = +∞ From (5), we get Proposition 4.3 A generalization of the J'-independence property between two fuzzy set is