In this note, we give four versions of Egoroff’s theorem in non-additive measure theory by using the condition (E) and the pseudo-condition (E) of set function and the duality relations between the conditions. These conditions employed are not only sufficient, but also necessary for the four kinds of Egoroff’s theorem, respectively.

Over the past few decades, numerous optimization-based methods have been proposed for solving the classification problem in data mining. Classic optimization-based methods do not consider attribute interactions toward classification. Thus, a novel learning machine is needed to provide a better understanding on the nature of classification when the interaction among contributions from various at...

in this paper, some results of the chebyshev type integral inequality for the pseudo-integral are proven. the obtained results, are related to the measure of a level set of the maximum and the sum of two non-negative integrable functions. finally, we applied our results  to the case of comonotone functions.

In 1974, Sugeno introduced the notion of fuzzy measure and integral to evaluate nonadditive or non-linear quality in systems engineering. In the same year, Dobrakov independently introduced the notion of submeasure from mathematical point of view to show that most of the theory of countably additive measures remain valid for such measures. Fuzzy measures and submeasures are both special kinds o...

Traditional probabilistic description of uncertainty is based on additive probability measures. To describe nonprobabilistic uncertainty, it is therefore reasonable to consider non-additive measures. An important class of non-additive measures are possibility measures, for which μ(A ∪ B) = max(μ(A), μ(B)). In this paper, we show that possibility measures are, in some sense, universal approximat...

The duality of truth and falsity in a Boolean algebra of propositions is used to generate a duality of belief and disbelief. To each additive probability measure that represents belief there corresponds a dual additive measure that represents disbelief. The dual measure has its own peculiar calculus, in which, for example, measures are added when propositions are combined under conjunction. A V...

A problem which is appealing to the intuition in view of the relative frequency interpretation of probability is to define a measure on a countable space which assigns to each point the measure 0. Such a measure of course becomes trivial if it is countably additive. Finitely additive measures of this type have been discussed by R. C. Buck [l] and by E. F. Buck and R. C. Buck [2]. In a discussio...

the stability problem of the functional equation was conjectured by ulam and was solved by hyers in the case of additive mapping. baker et al. investigated the superstability of the functional equation from a vector space to real numbers.in this paper, we exhibit the superstability of $m$-additive maps on complete non--archimedean spaces via a fixed point method raised by diaz and margolis.