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.
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In this paper we deal with continuous convergence and some related properties of sequences of functions. We present some conditions to get uniform convergence of the sequence involved to a constant function. As an application, we give a result on equivalence between modes of convergence.
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Absolute continuity, singularity and Lebesgue decompositions are studied, in the context of the so-called RD-convergence, for finitely and/or σ-additive measures taking values in super-Dedekind complete l-groups. By means of a Vitali-Hahn-Saks-Nikodým result, found in [4], we deduce a convergence theorem for the Lebesgue decompositions of an RD-convergent sequence of finitely additive measures.
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