Monotone Set Function Research Articles

Hardy–Littlewoodʼs inequalities in the case of a capacity

Hardy–Littlewoodʼs inequalities, well known in the case of a probability measure, are extended to the case of a monotone (but not necessarily additive) set function, called a capacity. The upper inequality is established in the case of a capacity assumed to be continuous and submodular, the lower — under assumptions of continuity and supermodularity.

Set-Valued Lebesgue and Riesz Type Theorems

Abstract In this paper, we continue a previous study concerning different types of pseudo-convergences of sequences of measurable functions with respect to set-valued nonadditive monotonic set functions and we establish some pseudo-versions of Lebesgue and Riesz theorems in the set-valued case. We also characterize some important structural properties of fuzzy multimeasures.

Pseudo-Convergences of Sequences of Measurable Functions on Monotone Multimeasure Spaces

In this paper, we study different types of pseudo-convergences of se- quences of measurable functions with respect to set-valued non-additive monotonic set functions and we establish some pseudo-versions of Egoroff theorem in the set-valued case. We also characterize important structural properties of monotone multimeasures. Mathematics Subject Classication

Asymmetric integral as a limit of generated Choquet integrals based on absolutely monotone real set functions

We introduce a generated Choquet integral with respect to absolutely monotone and sign stable set functions. Another type of integrals defined with respect to such a set function is obtained as a limit of an appropriate sequence of generated Choquet integrals.

Characterization for entropy and solution of set function on lattice

We propose a definition for the entropy of a monotone set function defined on a lattice which are not necessarily the whole power set, but satisfy the condition of regularity. Our definition encompasses the classical definition of Shannon for probability measures, as well as the definition of Marichal for classical fuzzy measures and may have applicability to most fuzzy measures which appear in applications. We give also an axiomatization of this entropy. This axiomatization is in the spirit of Faddeev's axiomatization for the classical Shannon entropy. After that, using same idea we introduce a generalization of the Shapley value for a set function defined on a lattice and give two types of necessary and sufficient conditions.

쇼케이 적분과 구간치 필요측도

Y. R<TEX>$\acute{e}$</TEX>ball<TEX>$\acute{e}$</TEX> [11]교수는 쇼케이적분 기준에 의한 필요측도의 표현에 관해 조사한다. 또한 쇼케이적분 표현관 관련된 필요측도의 순위를 결정 연장을 생각한다. 이 논문에서, 우리는 결정연장이 쇼케이 기대효용에 따른 애매한(구간치로 명명함) 필요측도를 가지는 경우를 생각한다. 더욱이, 구간치 필요측도에 대한 단조 집합치 함수를 갖는 기호에 대한 약 쇼케이적분표현과 필요측도에 대한 구간치 효용함수를 갖는 기호에 대한 강 쇼케이적분 표현에 대한 두 가지 정리를 증명한다. Y. R<TEX>$\acute{e}$</TEX>ball<TEX>$\acute{e}$</TEX> [11] discussed the representation of necessity measure through the Choquet integral criterian. He also consider a decision maker who ranks necessity measures related with Choquet integral representation. In this paper, we consider a decision maker have an "ambiguity"(say, interval-valued) necessity measure according to their Choquet's expected utility. Furthermore, we prove two theorems which are weak Choquet integral representation of preferences with a monotone set function for interval-valued necessity measures and strong Choquet integral representation of preferences with an interval-valued utility function for necessity measures.

구간치 쇼케이적분에 의해 정의된 단조 구간치 집합함수의 구조적 성질에 관한 연구

We introduce nonnegative interval-valued set functions and nonnegative measurable interval-valued Junctions. Then the interval-valued Choquet integral determines a new nonnegative monotone interval-valued set function which is a generalized concept of monotone set function defined by Choquet integral in [17]. We also obtained absolutely continuity of them in [9]. In this paper, we investigate some characterizations of the monotone interval-valued set function defined by the interval-valued Choquet integral, and such as subadditivity, superadditivity, null-additivity, converse-null-additivity.

THE AUTOCONTINUITY OF MONOTONE INTERVAL-VALUED SET FUNCTIONS DEFINED BY THE INTERVAL-VALUED CHOQUET INTEGRAL

In a previous work [18], the authors investigated autocontinuity, converse-autocontinuity, uniformly autocontinuity, uniformly converse-autocontinuity, and fuzzy multiplicativity of monotone set function defined by Choquet integral([3,4,13,14,15]) instead of fuzzy integral([16,17]). We consider nonnegative monotone interval-valued set functions and nonnegative measurable interval-valued functions. Then the interval-valued Choquet integral determines a new nonnegative monotone interval-valued set function which is a generalized concept of monotone set function defined by Choquet integral in [18]. These integrals, which can be regarded as interval-valued aggregation operators, have been used in [10,11,12,19,20]. In this paper, we investigate some characterizations of monotone interval-valued set functions defined by the interval-valued Choquet integral such as autocontinuity, converse-autocontinuity, uniform autocontinuity, uniform converse-autocontinuity, and fuzzy multiplicativity.

단조집합함수에 의해 정의된 구간치 쇼케이적분에 대한 르베그형태 정리에 관한 연구

In this paper, we consider Lebesgue-type theorems in non-additive measure theory and then investigate interval valued Choquet integrals and interval-valued fuzzy integral with respect to a additive monotone set function. Furthermore, we discuss the equivalence among the Lebesgue`s theorems, the monotone convergence theorems of interval-valued fuzzy integrals with respect to a monotone set function and find some sufficient condition that the monotone convergence theorem of interval-valued Choquet integrals with respect to a monotone set function holds.

Extensions of (weakly) null-additive, monotone set functions from rings of subsets to generated algebras

This paper shows that the greatest and least monotone extensions of a null-additive (resp. weakly null-additive), monotone set function from a ring of subsets to the algebra generated by the ring are null-additive (resp. weakly null-additive). In addition, the paper characterizes all the (weakly) null-additive, monotone extensions.

Entropy of capacities on lattices and set systems

We propose a definition for the entropy of capacities defined on lattices. Classical capacities are monotone set functions and can be seen as a generalization of probability measures. Capacities on lattices address the general case where the family of subsets is not necessarily the Boolean lattice of all subsets. Our definition encompasses the classical definition of Shannon for probability measures, as well as the entropy of Marichal defined for classical capacities. Some properties and examples are given.

On finite-state approximants for probabilistic computation tree logic

We generalize the familiar semantics for probabilistic computation tree logic from finite-state to infinite-state labelled Markov chains such that formulas are interpreted as measurable sets. Then we show how to synthesize finite-state abstractions which are sound for full probabilistic computation tree logic and in which measures are approximated by monotone set functions. This synthesis of sound finite-state approximants also applies to finite-state systems and is a probabilistic analogue of predicate abstraction. Sufficient and always realizable conditions are identified for obtaining optimal such abstractions for probabilistic propositional modal logic.

Order continuous of monotone set function and convergence of measurable functions sequence

In this paper, we introduce the concept of strongly order continuity of a set function, and use it to investigate the convergence of measurable functions sequence and of integral sequence on fuzzy measure space. Several equivalent conditions of strongly order continuity are presented. It is shown that a necessary and sufficient condition that Lebesgue's theorem remains valid for a monotone set function is that the set function possesses strongly order continuity.

Determining nonnegative monotone set functions based on Sugeno's integral: an application of genetic algorithms

Regarding the set of all information sources as the universe of discourse, we used a nonnegative monotone set function defined on its power set to describe the importance of each individual information source and their varied combinations. Such a set function is called an importance measure or a fuzzy measure. The Sugeno integral with respect to the nonnegative monotone set function possesses many desired properties, such as the fuzzy linearity when the set function is fuzzy additive, and can be adopted as an aggregation means in information fusion. In real problems, viewing the Sugeno integral as a multi-input single-output system, we use a genetic algorithm to determine the values of the importance measure from the input–output data of the system.

WEAK CONVERGENCE OF BOUNDED, MONOTONE SET FUNCTIONS IN AN ABSTRACT SETTING

We introduce an abstract treatment of the weak convergence for bounded monotone set functions which allows us to obtain some basic results generalizing well known theorems regarding classical weak and vague convergence and weak convergence of masses on normal topological spaces (e.g. Portmanteau type theorems, Direct and Converse Prokhorov type theorems). Moreover, we introduce a suitable topology (called the L\'evy-topology) in order to study the properties of this abstract convergence from a topological point of view.\newline

Using genetic algorithms to determine nonnegative monotone set functions for information fusion in environments with random perturbation

Due to some inherent interactions among diverse information sources, the classical weighted average method is not adequate for information fusion in many real problems. To describe the interactions, an intuitive and effective way is to use an appropriate nonadditive set function. Instead of the weighted average method, which is essentially the Lebesgue integral, we should thus use the Choquet integral or some other nonlinear integrals. To apply this alternative, more realistic approach to information fusion, we need to determine the nonadditive set function from given input-output data, viewing the nonlinear integral as a multi-input one-output system. In this paper, we employ an adaptive genetic algorithm to construct an approximate optimal nonnegative monotone set function from given input-output data in an environment with random perturbation. An example for diverse strengths of random perturbation is shown to demonstrate the efficiency of this algorithm.

CONVERGENCE THEOREMS FOR SEQUENCES OF CHOQUET INTEGRALS

The Choquet integral with respect to nonadditive monotone set functions, including imprecise probabilities and fuzzy measures, is a generalization of the classical Lebesgue integral. It is one kind of nonlinear functionals defined on a subspace of all real valued measurable functions. In this paper, several different types of convergence, including the mean convergence that is based on the Choquet integral, for sequences of measurable functions are considered, and the corresponding convergence theorems for sequence of Choquel integrals are demonstrated. Particularly, the theorem of convergence in measure is presented in a form of “necessary and sufficient condition” by using the structural characteristics of nonnegative monotone set functions. As an application of convergence theorems, the stability of a class of nonlinear integral systems is discussed.

PFB-Integrals and PFA-Integrals with Respect to Monotone Set Functions

A pair of new integrals, a PFB-integral and a PFA-integral, are introduced. Given a measurable space, these integrals are defined for nonnegative functions with respect to monotone set functions, and they are generalizations of both the Lebesgue integral and the fuzzy integral. Some primary properties of the PFB-integral and PFA-integral are discussed and an inequality between them is established in this paper.

Choquet integrals and natural extensions of lower probabilities

Both the Choquet integral with respect to monotone set functions and the natural extensions of lower probabilities are generalizations of the Lebesgue integral with respect to σ-additive measures. The relation between these generalizations is investigated. We show that the Choquet integral with respect to a belief measure is always greater than or equal to the corresponding natural extension. Also, we compare some concepts introduced in the theory of imprecise probabilities with concepts established in fuzzy measure theory, capacity theory, and Dempster-Shafer theory.

PAN-INTEGRALS WITH RESPECT TO IMPRECISE PROBABILITIES

As a unified generalization of both the Lebesgue integral and the Sugeno integral, pan-integrals were introduced for nonnegative measurable functions with respect to fuzzy measures. In this paper, the concept of a pan-integral is generalized for nonnegative measurable functions with respect to monotone set functions, which include coherent imprecise probabilities. An inequality is established between two functionals on the class of all nonnegative bounded measurable functions, the natural extension of a monotone lower probability and the corresponding pan-integral whose operators are the common addition and common multiplication. We also discuss the necessary and sufficient conditions for the equality between these two functionals. By using a new concept of para-additivity, we derive a necessary and sufficient condition for the equality when the lower probability is restricted to be a belief measure.