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

Convergence of sequence of measurable functions on fuzzy measure spaces

Fundamental convergence of sequences of measurable functions on fuzzy measure space

The extension theorem of F-measure spaces

The autocontinuity of set function and the fuzzy integral

By analogy with the classical case of a probability measure, we extend the notion of increasing convex (concave) stochastic dominance relation to the case of a normalized monotone (but not necessarily additive) set function also called a capacity. We give different characterizations of this relation establishing a link to the notions of distribution function and quantile function with respect to the given capacity. The Choquet integral is extensively used as a tool. In the second part of the paper, we give an application to a financial optimization problem whose constraints are expressed by means of the increasing convex stochastic dominance relation with respect to a capacity. The problem is solved by using, among other tools, a result established in our previous work, namely a new version of the classical upper (resp. lower) Hardy–Littlewood's inequality generalized to the case of a continuous from below concave (resp. convex) capacity. The value function of the optimization problem is interpreted in terms of risk measures (or premium principles).

In this article, we present and investigate new type of sequence spaces called by almost convergent Motzkin sequence spaces. We demonstrate that the newly introduced spaces are linearly isomorphic to the spaces of all almost convergent sequences and compute the ?-dual. Additionally, we characterize (M, Z) and (Z, M) for any given sequence space Z, and also determine the necessary and sufficient condition on a matrix P such that for every bounded sequence u, BM-core(Pu) ? K-core(u) and BM-core(Pu) ? st-core(u).

For sequences of measurable functions on a set-valued fuzzy measure space, the concepts of pseudo almost everywhere convergence, pseudo almost uniformly convergence, and pseudo-convergence in measure are introduced. Then, Egoroff’s theorem, Lebesgue’s theorem, and Riesz’s theorem are generalized from real-valued fuzzy measure spaces onto set-valued fuzzy measure spaces.

We define weighted geometric mean method of convergence for sequences in by using multiplicative calculus and obtain necessary and sufficient conditions under which convergence of sequences in follows from convergence of their weighted geometric means. We also obtain multiplicative analogues of Schmidt type slow oscillation condition and Landau type two-sided condition for the convergence in particular. Besides, we introduce the concepts of convergence, convergence, convergence, convergence for sequences of intuitionistic fuzzy numbers (IFNs) and apply the aforementioned conditions to achieve convergence in intuitionistic fuzzy number space. Examples of sequences are also given to illustrate the proposed methods of convergence.

Lebesgue theorems in non-additive measure theory

Statistical learning theory is regarded as an appropriate theory to deal with learning problems on small samples, and it has now become a novel research interest of the machine learning field. However, the theory is based on real-valued random samples and established on probability measure space; it rarely deals with learning problems based on fuzzy random samples and established on Sugeno measure space. It is well known that fuzzy random samples and Sugeno measure space are interesting and important extensions of real-valued random samples and probability measure space, respectively. Therefore, the statistical learning theory based on fuzzy random samples in Sugeno measure space is further discussed in this paper. Firstly, based on definitions of the distribution function and the expected value of fuzzy random variables in Sugeno measure space, the Hoeffding inequality of fuzzy random variables is proved. Secondly, for the sake of completeness of the paper, the key theorem of learning theory based on fuzzy random samples in Sugeno measure space is introduced. Finally, the bounds on the rate of uniform convergence of a learning process based on fuzzy random samples in Sugeno measure space are constructed.

In this study, first, lacunary convergence of double sequences is introduced in fuzzy normed spaces, and basic definitions and theorems about lacunary convergence for double sequences are given in fuzzy normed spaces. Then, we introduce the concept of lacunary ideal convergence of double sequences in fuzzy normed spaces, and the relation between lacunary convergence and lacunary ideal convergence is investigated for double sequences in fuzzy normed spaces. Finally, in fuzzy normed spaces, we give the concept of limit point and cluster point for double sequences, and the theorems related to these concepts are given.

The Application of Conflict Measure to Estimating Incoherence of Analyst's Forecasts about the Cost of Shares of Russian Companies

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.

The Schreier families are a collection of finite subsets of N and have been used to provide refinements of the following Banach space notions: unconditional basic sequences, convergent sequences, spreading model and strictly singular operators to name a few. We use the Schreier families to study subclasses of strictly singular operators on Banach spaces. We also provide a sufficient condition on the strictly singular operators implying every operator falls into one of these subclasses.

Characterizations of endograph metric and Γ-convergence on fuzzy sets