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

Boccuto, Antonıo; Dimitriou, Xenofon; Papanastassiou, N.

doi:10.2478/s12175-012-0070-5

Cited by 21 publications

(38 citation statements)

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“…This concept has been studied even in topological spaces (see e.g. Das, 2012;Das & Ghosal, 2011;Lahiri & Das, 2005;Lahiri & Das, 2007/2008 and in ( )-groups (see also , 2013bBoccuto, Dimitriou, & Papanastassiou, 2010b, 2012a, 2012b, 2012dBoccuto, Dimitriou, Papanastassiou, & Wilczyński, 2012, 2013. This notion has been several applications in the very recent literature.…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

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

Boccuto¹,

Dimitriou²

2013

JMR

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“…(d) A free filter F of N is said to be diagonal if and only if for every sequence (A n ) n in F and for each I ∈ F * there exists a set J ⊂ I, J ∈ F * such that J \ A n is finite for all n ∈ N (see also [6], [9]). …”

Section: ò ø óò× 21ºmentioning

confidence: 99%

“…A comprehensive survey can be found in [6]. There are also several versions of theorems of this kind for finitely or countably additive measures in the setting of filter convergence (for a related literature, see also [5][6][7][8][9], [14]). In [7], some Brooks-Jewett, Nikodým and Vitali-Hahn-Saks--type theorems are proved for positive and finitely additive lattice group-valued measures with respect to filter convergence, in which the pointwise convergence of the involved measures is required, not necessarily with respect to a single order sequence or regulator.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, for technical reasons, in our context we deal with (D)-convergence, because it is possible to use the Fremlin's lemma which allows to replace a series of (D)-sequences with a single regulator. In [5], [6], [8], [9], [14], some limit theorems are proved for finitely additive and not necessarily positive lattice group-valued measures, for diagonal filters which satisfy some additional properties. Finally, we pose some open problems.…”

Section: Introductionmentioning

confidence: 99%

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