1. Introduction
In these last years, on crisp setting, measures of information J, have been studied by many authors [ 1 ] [ 2 - 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 - 8 ]. Analogous studies were presented in [ 9 ] on fuzzy setting.
In order to study the integration in information theory without probability, in [ 10 - 12 ] the families
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 - 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
. For this reason, the families mentioned above will be adapted to the fuzzy setting and the corresponding class
will be introduced.
Example: let A be the fuzzy set of old men,
the fuzzy set of those old men, who seem ill, who are ill, who are seriously ill, respectively. These different conditions
are the conditioning variable events. The conditional information:
measure the influence of the grade of the illness in the old men.
The paper is organized in the following way: in Sect. 2 some preliminaires are recalled; in Sect. 3 the 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.
2. 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
is a measurable space.
Definition 2.1 In the fuzzy setting measure of the general information is a map
such that
:
(i)
(ii)
Following the idea presented in [ 10 , 12 ], assigned an information measure J, the following families are introduced:
(1)
(2)
The family (1) is not empty because it contains the whole set X and all supersets
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
:
is not an filter [ 21 , 22 ] because it is not stable with respect to the intersection between fuzzy sets.
3. 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
, conditioned by a variable event
is a map
such that
(j)
(jj)
From the previous axioms, it follows that
The condition (j) is the monotonicity, the (jj) means that all null sets
don’t condition any fuzzy set
.
4. Statement of the Problem
Taking into account the previous axiomatic statement, fixed an information measure J on
and any fuzzy set
, some classes of measures
, will be sought by supposing that
depends only on
and
. Now it is necessary to specify the family where H belongs.
Fixed A, our definition is restricted to the following family
at least
contains the whole space X, so this family is not empty. The condition
ensures that
: in fact if
from the monotonicity of J, it is
.
So, the information
is the function
expressed by a function
, such that
with
This justifies the domain of the function
.
From (3), (j), (jj),
the function
shall satisfy the following properties:
(I)
(II)
Setting
,
with
and
,
the following system of functional equations is obtained:
5. Solution of the problem
A function
continuous defined in the following set:
(3)
will be sought as an universal law in the sense that the equation and the inequality about the function of the system must be satisfied for all values and variables in their proper space, which satisfies the system [(1)-(2)].
Now, the following results are proved:
Proposition 4.1 A class of solutions of the system [(1)-(2)] is:
(4)
where
is any continuous function, strictly increasing with
Proof. The condition (1) follows by the monotonicity of the function h. The second one (2) results from the value
and the property of the function h.
Proposition 4.2 A class of solutions of the system [(1)-(2)] is:
(5)
where
is any continuous function, strictly increasing with
Proof. The proof is immediate. ,
Proposition 4.3 A class of solutions of the system [(1)-(2)] is:
(6)
where
is any continuous function, strictly increasing with
Proof. The proof is immediate. ,
From (4) (5) and (6), the following expressions of conditional information have been obtained, respectively:
(7)
(8)
where
is any continuous function, strictly increasing with
and
(9)
6. Conclusions
In this paper, for the first time, we present an axiomatic definition of the information
, when the conditioning event is variable.
We think that this axiomatic approach could be useful for future applications.
Acknowledgments
For the first author, this research is in the framework of GNFM of MIUR (Italy) and University “Sapienza” Roma.
The authors declare that they have no conflict of interest.