Dieudonné-type theorems for lattice group-valued kk-triangular set functions

summary:Some versions of Dieudonné-type convergence and uniform boundedness theorems are proved, for $k$-triangular and regular lattice group-valued set functions. We use sliding hump techniques and direct methods. We extend earlier results, proved in the real case. Furthermore, we pose some open problems

Rates of approximation for general sampling-type operators in the setting of filter convergence

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ideal limit theorems and their equivalence in ell group setting

We prove some equivalence results between limit theorems for sequences of $(\ell)$-group-valued measures, with respect to order ideal convergence. A fundamental role is played by the tool of uniform ideal exhaustiveness of a measure sequence already introduced for the real case or more generally for the Banach space case in our recent papers, to get some results on uniform strong boundedness and uniform countable additivity. We consider both the case in which strong boundedness, countable additivity and the related concepts are formulated with respect to a common order sequence and the context in which these notions are given in a classical like setting, that is not necessarily with respect to a same $(O)$-sequence. We show that, in general, uniform ideal exhaustiveness cannot be omitted. Finally we pose some open problems

Modes of Ideal Continuity of (l)-Group-Valued Measures

In this paper we deal with (ideal) continuity of lattice group-valued finitely additive measures, and prove some basic properties and comparison results. We investigate the relations between different modes of ideal continuity, and give some characterization. Finally we pose some open problems

Rates of approximation for general sampling-type operators in the setting of filter convergence

We investigate the order of approximation of a real-valued function f by means of suitable families of sampling type operators, which include both discrete and integral ones. We give a unified approach, by means of which it is possible to consider several kinds of classical operators, for instance Urysohn integral operators, in particular Mellin-type convolution integrals, and generalized sampling series. We deal with filter convergence, obtaining proper extensions of classical results

Abstract Korovkin-type theorems in modular spaces and applications

AbstractWe prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. Finally, we give some examples and applications to moment and bivariate Kantorovich-type operators, showing that our results are proper extensions of the corresponding classical ones

Abstract Theorems on Exchange of Limits and Preservation of (Semi)continuity of Functions and Measures in the Filter Convergence Setting

We give necessary and sufficient conditions for exchange of limits of double-indexed families, taking values in sets endowed with an abstract structure of convergence, and for preservation of continuity or semicontinuity of the limit family, with respect to filter convergence. As a consequence, we give some filter limit theorems and some characterization of continuity and semicontinuity of the limit of a pointwise convergent family of set functions. Furthermore, we pose some open problems

Convergences of sequences of measures and of measurable functions

In this dissertation we introduce new notions of convergence of measure sequences and using them we strengthen and extend classical limit theorems of Measure Theory like the convergence theorems of Nikodým, Brooks-Jewett, Vitali-Hahn-Saks and Schur. Moreover we define and study new notions of convergence for sequences of measurable functions which are weaker than the classical convergence in measure. Using these we prove new density theorems in ℝ. We consider the modes of convergence mentioned in the dissertation to be useful in the research in Probability Theory (e.g. in the study of ultra-weak laws of large numbers), in Topology (e.g. in the study of density topologies) and in Functional Analysis (e.g. in the study of function spaces of Baire-type and Ascoli-type theorems).Στην διατριβή αυτή εισάγονται νέες έννοιες σύγκλισης ακολουθιών μέτρων και με τη βοήθεια αυτών ισχυροποιούνται και επεκτείνονται κλασικά οριακά θεωρήματα της Θεωρίας Μέτρου όπως τα θεωρήματα σύγκλισης Nikodým, Brooks-Jewett, Vitali-Hahn-Saks και Schur. Επίσης ορίζονται και μελετούνται νέες έννοιες σύγκλισης για ακολουθίες μετρησίμων συναρτήσεων που είναι ασθενέστερες της κλασικής σύγκλισης κατά μέτρο. Με τη βοήθεια αυτών των συγκλίσεων αποδεικνύονται θεωρήματα πυκνότητας στο ℝ. Οι τρόποι σύγκλισης που αναφέρονται στην διατριβή θεωρούμε ότι μπορούν να είναι πρόσφορες στην έρευνα της Θεωρίας Πιθανοτήτων (π.χ. με τη μελέτη υπερασθενών νόμων των μεγάλων αριθμών), της Τοπολογίας (π.χ. με τη μελέτη τοπολογιών πυκνότητας) και της Συναρτησιακής Ανάλυσης (π.χ. με τη μελέτη χώρων συναρτήσεων τύπου Baire και θεωρημάτων τύπου Ascoli)

Convergence theorems for lattice group-valued measures

Convergence Theorems for Lattice Group-valued Measures explains limit and boundedness theorems for measures taking values in abstract structures. The eBook begins with a historical survey about these topics since the beginning of the last century, moving on to basic notions and preliminaries on filters/ideals, lattice groups, measures and tools which are featured in the rest of this text. Readers will also find a survey on recent classical results about limit, boundedness and extension theorems for lattice group-valued measures followed by information about recent developments on these kinds

Modular Filter Convergence Theorems for Urysohn Integral Operators and Applications

We prove some versions of modular convergence theorems for nonlinear Urysohn-type integral operators with respect to filter convergence. We consider pointwise filter convergence of functions giving also some applications to linear and nonlinear Mellin operators. We show that our results are strict extensions of the classical ones