Fuzzy Measure Space Research Articles

The structural characteristics of the fuzzy number-valued fuzzy measure on the fuzzy σ-algebra and their applications

In this paper, some structural characteristics of the fuzzy number-valued fuzzy measure on the fuzzy σ-algebra are discussed. On the fuzzy number-valued fuzzy measure space, the concepts of ‘almost’ and ‘pseudo-almost’ are introduced, and Riesz's theorem, Lebesgue's theorem and Egoroff's theorem for sequences of fuzzy measurable functions and some convergence theorems for sequences of fuzzy number-valued fuzzy integrals on fuzzy sets are proved by some structural characteristics of the fuzzy number-valued fuzzy measure.

FUZZY QUANTUM MODELS

We present some models of quantum mechanics which use fuzzy set approaches: fuzzy measurable spaces and fuzzy quantum posets of two types. Also presented are compatibility problems, observables and their representations, functional calculus of observables, and states (= probability measures).

On the existence of probability measures on fuzzy measurable spaces

We present some classes of fuzzy measurable spaces which possess at least one probability measure.

FUZZY MEASURES AND FUZZY INTEGRALS: AN OVERVIEW

This paper provides an overview of fuzzy measures, fuzzy integration theories and Choquet's capacity theory. Belief, plausibility, and possibility measures are characterized as Choquet capacities and as fuzzy measures. The relationship between possibility measures, fuzzy sets, and approximate reasoning is established. Recent results on extensions of fuzzy measures, structural characteristics of fuzzy measures, and convergence of function sequences on fuzzy measure spaces are presented. Fuzzy measure integration concepts due to Sugeno and Choquet and their applications are discussed. An extensive list of references to the literature of fuzzy measures, Sugeno and Choquet integrals, fuzzy probabilities, fuzzy random variables, probabilistic sets, and random sets is provided. Applicalions discussed or referenced include information fusion, information retrieval, approximate reasoning, artificial intelligence, uncertainty theory, and control and decision theory.

The autocontinuity of set function and the fuzzy integral

The autocontinuity and some other concepts of a set function are introduced and Sugeno's fuzzy measure with some annexed conditions is studied. On a fuzzy measure space, Egoroffs theorem is proved. It is also proved that F-mean convergence is equivalent to convergence in measure for a sequence of measurable functions. Finally, some convergence theorems of sequence of fuzzy integrals are proved, especially, using the concept of autocontinuity, a necessary and sufficient condition for the convergence in measure theorem is given.

Asymptotic Structural Characteristics of Fuzzy Measure and their Applications

In the fuzzy measure theory, as Sugeno’s fuzzy measures lose the additivity in general, the concept “almost”, which is well known in the classical measure theory, splits into two different concepts “almost” and “pseudo-almost”. It complicates the relations among the convergences of the sequence of measurable functions on the fuzzy measure space, and increases the content of the theory of convergences of the seauence of fuzzy integrals. In order to replace the additivity, it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at the sequences of sets which are called “waxing” and “waning”, and to introduce some new concepts, such as “autocontinuity”, “converse-autocontinuity” and “pseudo-autocontinuity”. Ihey describe some asymptotic structural characteristics of a fuzzy measure. Bу means of them, we give four forms of generalization for each of the Egoroff’s theorem, the Liesz’s theorem and the Lebesgne’s theorem respectively, and prove a lot of convergence theorems on the sequence of fuzzy integrals.

The extension theorem of F-measure spaces

In this paper an important result is the extension theorem, where we find the conditions under which a fuzzy measurable space or a fuzzy measure space can be extended to a normal F-measurable space or to a normal F-measure space, respectively.