It is well known that the theory of probability can be treated and developed in a consistent and uniform way using the classical theory of measure and integration. Indeed, the Russian scientist Kolmogorov identified probability with normalized classical measures and used the Lebesgue theory of integration to give a logically consistent and unifying account of probability theory. In this paper, ...

We present a survey of various approaches for the refinement of the Sugeno integral, based on the lexicographic order or the principle of discarding common values between acts. We try to relate these different approaches and compare their merits.

We extend the concept of fuzzy integral based on ⊕-decomposable measure from nonnegative fuzzy measurable function to extended real-valued fuzzy measurable function. Further investigations of fuzzy integrals based on pseudo-additive decomposable measure are carried out. Meanwhile, the space (S(μ),σ(·,·)) of all fuzzy measurable function will be proved to be a pseudo-metric space. Finally, as an...

Abstract: Recently, the two most general and well-known fuzzy integrals (Sugeno's integral and Choquet's integral) were characterized. It was also shown how both integrals fitted the same formal model, and they only differed in the operators used. In this paper we try to analyze whether it is possible to improve those characterization theorems, and we also study the possibility of defining othe...

The paper deals with the aggregation of classification rules by means of fuzzy integrals, in particular with the fuzzy measures employed in that aggregation. It points out that the kinds of fuzzy measures commonly encountered in this context do not take into account the diversity of classification rules. As a remedy, a new kind of fuzzy measures is proposed, called similarity-aware measures, an...

Morphological tools can provide transformations suitable for real projective images, but the camera and objects to be analyzed have to be positioned in such a manner that a regular mesh on the objects projects a regular mesh on the image. A morphological modification of the image is thus the projection of an equivalent operation on the object. Otherwise, due to perspective effects, a morphologi...

The classical Jensen inequality for concave function φ is adapted for the Sugeno integral using the notion of the subdifferential. Some examples in the framework of the Lebesgue measure to illustrate the results are presented.

We are interested in representations and characterizations of lattice polynomial functions f : L → L, where L is a given bounded distributive lattice. In an earlier paper [4, 5], we investigated certain representations and provided various characterizations of these functions both as solutions of certain functional equations and in terms of necessary and sufficient conditions. In the present pa...

Fuzzy integral is an important tool to study fuzzy differential equations. Under normal circumstances, there are two basic limitations: limited of integral interval and boundedness of integrand. However, in practical problems, it is difficult to calculate when integral interval is not common interval. Then fuzzy integral on infinite interval is taken into consideration. In this paper, definitio...

Aggregation refers to the process of combining numerical values x1, . . . , xm into a single one M (x1, . . . , xm), so that the final result of aggregation takes into account all the individual values. In decision making, values to be aggregated are typically preference or satisfaction degrees and thus belong to the unit interval [0, 1]. This paper aims at investigating the Sugeno integral whi...