This chapter contains a historical survey about limit and boundedness theorems for measures since the beginning of the last century. In these kinds of theorems, there are two substantially different methods of proofs: the sliding hump technique and the use of the Baire category theorem. We deal with Vitali-Hahn-Saks, Brooks-Jewett, Nikodým convergence and boundedness theorems, and we consider also some related topics, among which Hahn-Schur-type theorems and some other kind of matrix theorems, the uniform boundedness principle and some (weak) compactness properties of spaces of measures. In this context, the Rosenthal lemma, the biting lemma and the Antosik-Mikusinski-type diagonal lemmas play an important role. We consider the historical evolution of convergence and boundedness theorems for σadditive, finitely additive and non-additive measures, not only real-valued and defined on σ-algebras, but also defined and/or with values in abstract structures.

Convergence Theorems for Lattice Group-Valued Measures

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.

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Ideal Limit Theorems and Their Equivalence in $(\ell)$-Group Setting

Some types of compactness in the ideal context are dened and relations between ideal exhaustiveness and equicontinuity of measures are investigated. As applications, some versions of limit theorems involving ideal pointwise convergence of measure sequences and some weak compactness results related to integral functionals are presented.

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Ideal Exhaustiveness, Weak Convergence and Weak Compactness in Banach Spaces

In this paper we introduce the I- and I*-convergence and divergence of nets in (l)-groups. We prove some theorems relating different types of convergence/divergence for nets in (l)-group setting, in relation with ideals. We consider both order and (D)-convergence. By using basic properties of order sequences, some fundamental properties, Cauchy-type characterizations and comparison results are derived. We prove that I*-convergence/divergence implies I-convergence/divergence for every ideal, admissible for the set of indexes with respect to which the net involved is directed, and we investigate a class of ideals for which the converse implication holds. Finally we pose some open problems.

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Ideal convergence and divergence of nets in $(\ell)$-groups

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.

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

We investigate fundamental properties of I-exhaustiveness and I-convergence of real-valued function sequences, giving some characterizations. Furthermore, we establish new versions of Ascoli and Helly theorems, giving also applications to measure theory. Finally, we pose an open problem.

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Ascoli-type theorems and ideal (α)-convergence

We introduce and study the notions of strong a-convergence, a stronger form of the known a-convergence (or continuous convergence), and of I-strong exhaustiveness, where I is an ideal of subsets of N, of a sequence of functions from a metric space (X,d) to another metric space (Y,�) and, among others, necessary and sufficient conditions for the continuity of the I-pointwise limit of a sequence of functions are derived.

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E. Athanassiadou

Papers

Ascoli-type theorems and ideal (α)-convergence

STRONG $\bbb{a}$-CONVERGENCE AND IDEAL STRONG EXHAUSTIVENESS OF SEQUENCES OF FUNCTIONS

Modes of (α)-convergence of sequences of functions