1. Introduction

Murofushi [1] had introduced the fuzzy measurement concept that is not monotone by removing monotonousness characteristics defined on fuzzy measurements and examined the features that belong to a special subset that has bounded variation feature of these measurements in his study. In Jang and Kwon’s [2] study, they had extended Murofushi’s assumptions with the help of the based on bounded variation that Schramn [3] had presented. Narukawa [4] had examined the features of convergence in Banach space that non-monotonic bounded variation structured measurements had generated. In this study, features of the cone structure created by the fuzzy measurements that had bounded variation feature had been examined and it had been indicated that it had the Riesz decomposition characteristic. Thus, it had been showed that every fuzzy measurement could be expressed as the combination of the non-monotonic characteristics.

2. Non-Monotonic Fuzzy Measure Spaces

Let X be defined as non-empty set and be defined as the measurable space for algebra that is described on this set.

Definition 2.1: If set function which is defined on measurable space provides the following characteristics, it is described as the fuzzy measurement.

Definition 2.2: If set function which is defined on measurable space provides the characteristic, it is described as the non-monotonic fuzzy measurement.

It is clear that has generated chain of under the coverage relation of the subsets belonging to it. Every sequential element of the chain has composed a relation and hereby chain structure has appeared as the combination of the sequential elements.

Definition 2.3: Total variation value that belongs to non-monotonic fuzzy measurement function defined on the measurable space is described for a linkage as

(1)

The decomposition of this expression is as follows:

(2)

expression for has been showed with and it is defined as the bounded variation for Let fuzzy measurements set be given as . The set of has a closed structure for every and it has generated linear space. Although the subset generates the positive cone structure in related linear space. If non-monotonic fuzzy measurement function has bounded variation, is the element of the linear space. The opposite of this suggestion is also true. (Aumann and Shapley, 1974:27)

We can describe the norm of defined on the linear space of It has been known that the inequality of is valid for [5].

Theorem 2.1: has generate the norm structure that is defined on the linear space and has the Banach space structure.

Proof: (Jang and Kwon, 1997:104,105) and (Schramn, 1985) had described the related proof for increasing convex functions series as the –bounded variation structure. Privately, if we describe the series for every as bounded variation structure is obtained.

It is clear that the cone of has the closed cone characteristic according to norm in the Banach space

Definition 2.4: Let be Banach space and be the closed cone defined in this space. Closed cone has the powerful Levi characteristics if every increasing sequence that is defined in provides norm convex feature for [6].

Theorem 2.2: Let be given for Banach space and be given for closed cone belonging to this space. If cone has powerful Levi characteristic, it has also the norm normality feature at the same time.

Definition 2.5: Let be linear space and be the cone in this space. If the equality of is valid, is defined as the generating cone. From the definition, it is clear that cone is the generating cone of linear space.

Let be given as the order relation (reflexive, transitive, anti-symmetric) on the linear space. When and were given, it is defined as the ordered vector space if the following conditions are satisfied.

It is clear that linear space has the ordered vector space characteristic.

Definition 2.6: Let be given in the (V,≥) ordered vector space. If this space has Riesz decomposition characteristic, there are such elements for the inequality can be written under the conditions of

When data features had been provided for the given two theorems Banach space and cone embedded into it in the rest of study, related space has showed that it satisfies the Riesz decomposition characteristic.

Theorem 2.3 [7] Let be Banach space and cone that is defined in this space be closed, generating and normal structured. Banach space should provide the following feature to have the Riesz decomposition characteristic.

Ÿ Let at most three vectors be for providing the and sequence. In this case, the elements for every can be found as satisfying the conditions under the assumption of

Theorem 2.4 [8] If is ordered Banach space and cone which is defined in this space is closed, generating, norm normal structured cone, it has the following characteristics.

Ÿ Banach space has the Riesz decomposition characteristic.

Ÿ The dual of Banach space is Riesz space.

Ÿ The dual of Banach space has the Riesz decomposition characteristic.

It had been showed that ordered Banach space and cone that belongs to this space have closed, generating and norm normal characteristics. With the help of the Theorem 2.3, it is clear that Banach space has Riesz decomposition characteristic so cone has it. This case has indicated that every fuzzy measurement can be expressed with the help of non-monotonic fuzzy measurements.

3. Conclusions

It had been revealed that the cone structure formed by fuzzy measurements which were defined in the ordered Banach space that non-monotonic fuzzy measurements created has the Riesz decomposition characteristic under the closed, generating and norm normal structured feature thus fuzzy measurements can be expressed as the linear combination of the non-monotonic fuzzy measurement.

Notes

1. The set defined on the linear space is described as the cone if it provides the following conditions.

2. norm that is defined on the Banach space is equivalent to the

3.

4. inequality is valid if