Uniform s-Boundedness and Convergence Results for Measures with Values in Complete l-Groups

Boccuto, Antonıo; Candeloro, Domenico

doi:10.1006/jmaa.2001.7715

Cited by 20 publications

(24 citation statements)

References 15 publications

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“…Note that, in general, these concepts are not identical. Indeed there exist bounded finitely additive lattice group-valued measures, which are not globally (s)-bounded (see Boccuto & Candeloro, 2002, Example 2.17), while every bounded finitely additive ( )-group-valued measure is (s)-bounded too (see Boccuto, Dimitriou, & Papanastassiou, 2010a, Theorem 3.1).…”

Section: Set Functions and Fn-topologiesmentioning

confidence: 99%

“…Indeed, in general, many problems involving finitely additive measures can be solved by finding suitable σ-additive measures, related to them, and then studying their properties. One of the main tools in this setting is the Stone extension, by means of which it is possible to construct a globally σ-additive measure, defined on a larger σ-algebra than the original one (see also Boccuto & Candeloro, 2002;Boccuto & Candeloro, 2004;Sikorski, 1964).…”

Section: This Proves ααα)mentioning

confidence: 99%

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

Boccuto¹,

Dimitriou²

2013

JMR

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We prove some equivalence results between limit theorems for sequences of ( )-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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Section: Set Functions and Fn-topologiesmentioning

confidence: 99%

Section: This Proves ααα)mentioning

confidence: 99%

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

Boccuto¹,

Dimitriou²

2013

JMR

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“…The wide literature originated by the classic Nikody'm and Vitali-Hahn-Saks theorems has produced interesting results in many abstract settings (see for instance [12,4,17,14]). In this framework recently also the so-called filter convergence has been considered, though in this case it is quite difficult to obtain general positive results.…”

Section: Introductionmentioning

confidence: 99%

Filter Convergence and Decompositions for Vector Lattice-Valued Measures

Candeloro¹,

Sambucini²

2014

Mediterr. J. Math.

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Filter convergence of vector lattice-valued measures is considered, in order to deduce theorems of convergence for their decompositions. First the σ-additive case is studied, without particular assumptions on the filter; later the finitely additive case is faced, first assuming uniform sboundedness (without restrictions on the filter), then relaxing this condition but imposing stronger properties on the filter. In order to obtain the last results, a Schur-type convergence theorem, obtained in [8], is used.2010 AMS Mathematics Subject Classification: 28B15, 28B05, 06A06, 54F05.

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“…For classical results on such theorems in the context of (l)-groups see, for instance, [5], [7]. Note that the ideal I d of the subsets of the natural numbers having zero asymptotic density is a P -ideal (see [12], [15]), and that I d -convergence coincides with the statistical convergence (see [13]).…”

Section: Introductionmentioning

confidence: 99%