N. Papanastassiou

Schur lemma and limit theorems in lattice groups with respect to filters

Some Schur, Nikodým, Brooks-Jewett and Vitali-Hahn-Saks-type theorems for (ℓ)-group-valued measures are proved in the setting of filter convergence. Finally we pose an open problem.

Some versions of limit and Dieudonné-type theorems with respect to filter convergence for (ℓ)-group-valued measures

Some limit and Dieudonné-type theorems in the setting of (ℓ)-groups with respect to filter convergence are proved, extending earlier results.

Basic matrix theorems for

Some aspects of the theory of order and (D)-convergence in (ℓ)-groups with respect to ideals are investigated. Moreover some new Basic Matrix Theorems are proved.

\mathcal{I}

Abstract. In this paper we present some different types of ideal convergence/divergence and of ideal continuity for Riesz space-valued functions, and prove some basic properties and comparison results. We investigate the relations among different modes of ideal continuity and present a characterization of the (AP)-property for ideals of an abstract set Λ. Finally we pose some open problems.

-convergence in (ℓ)-groups

In the study by Papanastassiou and Papachristodoulos, 2009 the notion of 𝑝-convergence in measure was introduced. In a natural way 𝑝-convergence in measure induces an equivalence relation on the space 𝑀 of all sequences of measurable functions converging in measure to zero. We show that the quotient space ℳ is a complete but not compact metric space.

Modes of ideal continuity and the additive property in the Riesz space setting

For every ordinal ξ<ω1 we define a new type of convergence for sequences of functions (ξ-uniform pointwise) which is intermediate between uniform and pointwise convergence. Using this type of convergence we obtain an Egorov type theorem for sequences of measurable functions.