Pseudo-Analysis: measures of general conditional information

A R T I C L E I N F O A B S T R A C T Article history: Received: 28 December, 2016 Accepted: 06 February, 2017 Online: 25 February, 2017 The aim of this paper is to continue our study of information in the setting of Pseudo-Analysis. We shall present, by axiomatic way, the definition of measures of general conditional information and we shall study particular measure by using a system of functional equations in which it is present a pseudo-operation. We know that J.Aczel is the founder of the Theory of Functional Equations and he solved the so called ”Cauchy Equation”. The method used in this paper consists in reducing the principal equation, to some basic known equations solved by Aczel and his school. With Benvenuti we studied a generalization of the Cauchy Equation and following these our results, we are able to give the general solution of the system and the expression for this measure of general conditional information.


Introduction
Since 1967 Kampé De Feriét and Forte introduced, by axiomatic way, the definition of measures J for general information, where general means that J is defined without probability [1].
Later, in [2], we introduced some particular family of crisp set (N , F , I ∞ , I 0 ), in order to study the integration in information theory without probability. We have used them for the definition of measures of general conditional information [3] .
In this paper, we would continue the researches in the setting of Pseudo-Analysis, started in [4], by using pseudo-addition and pseudo-difference. In particular we shall study measures for general conditional information for crisp sets.
The properties of the form of conditional information have translated in a system of functional equations [5], for which we shall give a class of solutions.
Moreover, by using the property of Jindependence we obtain another equation: we shall find the general solution through our previous result [6].
The paper is organize in the following way: in Sect.2 we recall some preliminaires; in Sect.3 we give the definition of general information conditioned by a variable event in pseudo-analysis. In Sect.4 we con-sider the statement of the problem and we traslate the properties of the form of conditional information in a system of functional equations.
We shall distinguish two cases: general case and independent case. In this last case the definition of independence is given by using the pseudo-analysis. We shall show some classes of solutions in Sect. 5. Sect.6 is devoted to the conclusions.

Preliminary notations 2.1 Pseudo-operations
We follow the Theory of Pseudo-Analysis introduced by E.Pap and his collegue [7], which consider the definition of pseudo operations. In particular, we shall use the pseudo-addition ⊕ and the pseudo-difference : for knownledge about these pseudo-operations, we refer to [8]. which is commutative, associative, strictly increasing with respect ≤, with 0 as neutral element.
We shall consider only particular operation ⊕ expressed by a function g, called generator function in the following way: where g : [0, M] −→ [0, +∞] and it is bijective, continuous and strictly increasing with g(0) = 0 and g(+∞) = +∞.
Definition 2.2 The pseudo-difference is a mapping, which we shall define through the same function g :

Measures of general information in classical analysis
Following [1], let X be an abstract space and A a σ −algebra of all subsets of X, such that (X, A) is a measurable space.

Definition 2.3 Measure of the general information is a mapping
Moreover, we have the following [1]: Definition 2.4 Given a subfamily K ⊂ A, two sets K, K ∈ K, K K , K ∩ K ∅ are called J-independent (i.e. independent with respect to J) if the couple (K, K ) satisfies the following: From [2], assigned an information measure J, we have considered the family: The family (4) is not empty because it contains the empty set ∅ and all subsets F of F ∈ I +∞ : I +∞ is not an filter [9] because it is not stable with respect to the intersection between fuzzy sets.
Given the family H = A − I +∞ we recall from [3],

Definition 2.5 The measure of general conditional information of any set A ∈ A conditioned by a fixed H ∈ H, (J(A|H)) is a mapping
Moreover, from [2], we have the following: conditioned by a fixed event H ∈ H if the couple (K, K ) satisfies the following condition: 3 Pseudo-analysis: measures of general conditional information In [4], for the first time, we have introduced the definition of J− independence property in the setting of pseudo-analysis, and we have used it to find the information of the union of two sets A, A ∈ A : J(A ∪ A ). From now on we consider a pseudo-addition ⊕ g generated by a function g as in (1).
Definition 3.1 Given a subfamily K ⊂ A, and a pseudo-addition ⊕ g , two sets K, K ∈ K , are called Jindependent in pseudo-analysis if the couple (K, K ) satisfies the following: In pseudo-analysis, for the general conditional information, we shall replace the common addition in (3) with the pseudo-addition ⊕ g , so we shall propose the following: Definition 3.2 Given a subfamily K ⊂ A, two sets K, K ∈ K are called J-conditional independent in pseudo-analysis if the couple (K, K ) satisfies the following property: In [10] we have generalize the property of J−independence.

Independent case
In this paragraph, we shall consider J− independent sets in pseudo-analysis.
We suppose that there exist two sets K, K ∈ K, K K , K ∩ K ∅, which are J− independent in pseudoanalysis in the sense of (6). From (7), and taking into account (8), it is On the other hand, by (8), Then, we obtain the condition of J−independence in pseudo-analysis: t t , t ≥ y, t ≥ y.

Solutions of the problem
Now, we are giving some solutions of the problem, distinguishing two previous cases.

Proposition 5.2 Another class of continuous solutions of the system
where µ is the product of the generator function g of the operation g given by (2) and m is any function as in (12).
Proof. Let m be a particular function solution of the system (e1) − (e3) as in Prop. [5.1], which defines a pseudo-addition ⊕ m . From (2) and (10), it is By the properties of g and m the solutions are continuous.
Fixed y = y * , the condition (e4) is setting the equation (15) becomes The equation (17) is a particular case of a general Cauchy equation on suitable hypothesys on the function F, when ⊕ is any pseudo-addition not necessary expressed by a generator function g. The equation (17) has been solved by Benvenuti and the authors in [6] in many general cases.
In particular, when this pseudo-addition is generated by function g, we found all continuous solutions. Here, we neglect trivial solutions and we consider only the most meanigfull solution. We recall the result from [6]:

Theorem 5.3 The solution of the general Cauchy equation
under suitable hypothesys on F, when the operation ⊕ g is defined by a generator function g is (really, we should say of a class of solution depending on a parameter λ) the following continuous function with g the generator function of the pseudo-addition (2).
By using the previuos result, the class of solution of (16) is with g the generator function of the pseudo-addition (2). Now, we can go back to our original problem concerning the J− independence property in pseudoanalysis and we are ready to give the main theorem.
Let L be any family of continuous function Λ : is the family, depending on any element λ(y) of L, Proof. Now, fixed any function Λ(y) ∈ L, we are verifying that (22) is solution of (e4): y). It is easy to see that any function (22) is continuous. Proof. The monotonicity of the function Φ Λ(y) (t) doesn't depend on the variable y; moreover Λ(y) is positive. Then, ∀ t ≤ t * and ∀ Λ(y) ∈ L, Φ Λ(y) (t) ≤ Φ Λ(y) (t ).
As conseguence of Theorem [5.4] and of Proposition [5.5], we get the following Theorem 5. 6 The only solution of the equations t, t ∈ [0, +∞], t t , t ≥ y, t ≥ y.
is the family, depending on any element of L, given by (21), Φ Λ(y) (t) = g −1   Λ(y) · g(t) It is easy to see that the conditions (e2) and (e3) are not compatible with the independent property. www.astesj.com 39

Conclusion
In this paper, given a measure of general information J, we have defined the measure of general conditional information J * (A|H) of any set A ∈ A conditioned to H ∈ H. Moreover, we have considered J * (A|H) depending only on J(A ∩ H) and J(H) through a function Φ. The properties of this J * (A|H) are translated in a system of functional equations.
In order to look for solutions of the system, we distinguish two cases: the first concerning monotonicity and particular values of J * (A|H), system (e1)-(e3), the second one related to monotonicity and independence property, equations (e1) and (e4).
I-General case: system (e1)-(e3) Some classes of the measure of general conditional information are: from (12)