General Information Conditioned by a Variable Event

The aim of this paper is to present, by axiomatic way, an idea about the general conditional information of a single, fixed fuzzy set when the conditioning fuzzy event is variable. The properties of this conditional information are translated in a system of functional equations. Some classes of solutions of this functional system have been found.


INTRODUCTION
In these last years, on crisp setting, measures of information J, have been studied by many authors [1] [ [2][3][4]. Later particular researches have been done to these information measures defined without probability. The last information measures are called general because they are defined without any probability [5][6][7][8]. Analogous studies were presented in [9] on fuzzy setting.
In order to study the integration in information theory without probability, in [10][11][12] the families 0 , , , ∞     have been introduced. The mentioned classes of crisp sets have replaced the families of null sets in the classical theory of integration (i.e. with respect to an additive measure or a probability). A detailed overview can be found in [13]. Later, since 2005, on fuzzy setting, measures of conditional information, when the conditional event is fixed, have been considered and studied (see [14][15][16]). Indeed, in this paper, a definition of the general conditional information of a fixed fuzzy set A, when the conditioning fuzzy set H is variable will be presented and it will be indicated by ( ) J A H . For this reason, the families mentioned above will be adapted to the fuzzy setting and the corresponding class A  will be introduced.
Example: let A be the fuzzy set of old men, 1 2 3 , , H H H the fuzzy set of those old men, who seem ill, who are ill, who are seriously ill, respectively. These different conditions 1 2 3 , , H H H are the conditioning variable events. The conditional information: ( ) ( ) ( ) definition of general information conditioned by a variable event is given. The statement of the problem is presented in Sect. 4 and in Sect. 5 the properties of the form of conditional information are translated in a system of functional equations [17], for which some classes of solutions are shown. Sect. 6 is devoted to the conclusions.

PRELIMINAIRES
In this paragraph, the definition of general information for fuzzy sets is recalled [9]. The concept of fuzzy set was introduced by Zadeh in [18], for all knowledge see [19,20].
Let X be an abstract space and  a-algebra of all fuzzy sets of X, such that ( ) , X  is a measurable space.
Definition 2.1 In the fuzzy setting measure of the general information is a map Following the idea presented in [10,12], assigned an information measure J, the following families are introduced: The family (1) is not empty because it contains the whole set X and all supersets N ′ of  is not an ideal [21,22] because it is not stable with respect to the union between fuzzy sets.
The family (2) is not empty because it contains the empty set ∅ and all subsets F' of F +∞ ∈  : is not an filter [21,22] because it is not stable with respect to the intersection between fuzzy sets.

MEASURE OF GENERAL CONDITIONAL INFORMATION BY A VARIABLE EVENT
From now on, the family +∞ = −    shall be considered and measure of general conditional information of a fixed fuzzy set ( ) defined on the family  will be introduced. Definition 3.1 Measure of general information of a fixed A∈  , conditioned by a variable event H ∈  is a map From the previous axioms, it follows that ( ) ( ). So, the information ( ) This justifies the domain of the function Φ . From (3), (j), (jj), 1 2 , , H H ∀ ∈ the function Φ shall satisfy the following properties: , , , , 0, x y y z z ∈ +∞ and 1 x z ≤ , , y z x z y z x t ≤ ≤ ≤ ≤ the following system of functional equations is obtained: , .
x y z x y z x z y z x z y z

SOLUTION OF THE PROBLEM
A function Φ continuous defined in the following set:

CONCLUSIONS
In this paper, for the first time, we present an axiomatic definition of the information ( ) J A H , when the conditioning event is variable.
We think that this axiomatic approach could be useful for future applications.