Ondrej Hutník

The smallest semicopula-based universal integrals I: Properties and characterizations

Abstract In this article we provide a detailed study of basic properties of the smallest universal integral I S with S being the underlying semicopula and review some of the recent developments in this direction. The class of integrals under study is also known under the name seminormed integrals and includes the well-known Sugeno and Shilkret integrals as special cases. We present some representations and equivalent formulations for I S . Further we discuss basic properties such as translatability, linearity, homogeneity, and maxitivity. In particular, we provide the characterizations of the two prominent integrals of Sugeno and Shilkret. We also raise and discuss some open questions.

The smallest semicopula-based universal integrals II

In this paper we continue studying the smallest universal integral I S having S as the underlying semicopula. We present convergence theorems for I S -integral sequences including monotone, almost everywhere, almost uniform, in measure and in mean converging sequences of measurable functions, respectively. It emerges that these convergences characterize the underlying measure properties such as null-additivity, monotone autocontinuity and autocontinuity. We provide many examples and counter-examples as well as a few interesting open problems.

On distance distribution functions-valued submeasures related to aggregation functions

Probabilistic submeasures generalizing the classical (numerical) submeasures are introduced and discussed in connection with some classes of aggregation functions. A special attention is paid to triangular norm-based probabilistic submeasures and more general semi-copula-based probabilistic submeasures. Some algebraic properties of classes of such submeasures are also studied.

ON A CERTAIN CLASS OF SUBMEASURES BASED ON TRIANGULAR NORMS

In this paper we study a generalization of a submeasure notion which is related to a probabilistic concept, especially to Menger spaces where triangular norms play a crucial role. The resulting notion of a τT-submeasure is suitable for modeling those situations in which we have only probabilistic information about the measure of the set. We characterize a class of universal τT-submeasures (i.e., τT-submeasures for an arbitrary t-norm T) and give explicit formulas for τT-submeasures for some classes of t-norms. Also, transformations and aggregations of τT-submeasures are discussed.

Beyond the scope of super level measures

Abstract We expand the theoretical background of the recently introduced outer measure spaces theory of Do and Thiele in harmonic and time-frequency analysis context. In the context of non-additive measures and integrals, we propose a certain framework for a natural extension of the basic ingredients of the theory (i.e., the concept of size, outer essential supremum and the corresponding super level measure) which, besides the covering the previously considered cases, permits us to introduce a further substantial extension of a class of non-additive integrals. All these notions are studied in detail and exemplified.

An integral with respect to probabilistic-valued decomposable measures

Abstract Several concepts of approximate reasoning in uncertainty processing are linked to the processing of distribution functions. In this paper we make use of probabilistic framework of approximate reasoning by proposing a Lebesgue-type approach to integration of non-negative real-valued functions with respect to probabilistic-valued decomposable (sub)measures. Basic properties of the corresponding probabilistic integral are investigated in detail. It is shown that certain properties, among them linearity and additivity, depend on the properties of the underlying triangle function providing (sub)additivity condition of the considered (sub)measure. It is demonstrated that the introduced integral brings a new tool in approximate reasoning and uncertainty processing with possible applications in several areas.

The smallest semicopula-based universal integrals III: Topology determined by the integral

Abstract Motivated by the recent results on topology determined by the prominent Sugeno and Choquet integrals, we study the topology on the space of measurable functions for a non-additive measure, which is determined by the universal integral based on a semicopula. We define a family of (pseudo-)metrics on the set of measurable functions parameterized by a semicopula and study their properties. We show that for a semicopula without zero divisors the convergence in the constructed (pseudo-)metric is equivalent to convergence in (non-additive) measure as well as other two types of convergences of measurable functions. For each such a (pseudo-)metric we construct a family of subsets of measurable functions and we provide sufficient conditions for this family to be a topology on the set of measurable functions. Moreover, we show that for some natural class of semicopulas the corresponding topological spaces are all equivalent, being equivalent to the topology determined by the Choquet integral.

Approximate continuity and topological Boolean algebras

Abstract In this paper we describe a net limit method for constructing topologies on given Boolean algebras. If functions in this limit procedure are monotone and approximately continuous in a generalized sense, then the obtained process is recursive.