Basic matrix theorems for $\mathcal{I}$-convergence in (ℓ)-groups

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

doi:10.2478/s12175-012-0053-6

Cited by 21 publications

(31 citation statements)

References 9 publications

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“…We now recall the concept of order ideal convergence (see also Boccuto, Dimitriou, & Papanastassiou, 2012a).…”

Section: Ideal Convergencementioning

confidence: 99%

“…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%

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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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“…We now recall the concept of order ideal convergence (see also Boccuto, Dimitriou, & Papanastassiou, 2012a).…”

Section: Ideal Convergencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Boccuto¹,

Dimitriou²

2013

JMR

View full text Add to dashboard Cite

show abstract

“…Examples of such spaces are R N and L 0 (X, S, µ) with µ positive, σ-additive and σ-finite (see also [8]). …”

Section: Limit Theoremsmentioning

confidence: 99%

“…The following concepts and results were given in [8]. A Dedekind complete (l)-group R is said to be super Dedekind complete if every subset R 1 ⊂ R, R 1 = ∅ bounded from above contains a countable subset having the same supremum as R 1 .…”

Section: Limit Theoremsmentioning

confidence: 99%

“…The theory of convergence with respect to ideals was introduced in [15] and is widely studied in the literature, in particular in problems concerning limit and integrals, even in abstract structures (see [8], where also a version of the basic matrix theorem is given in the context of (l)-groups). Observe that, in general, ideal convergence is strictly weaker than ordinary convergence (see [15]).…”

Section: Introductionmentioning

confidence: 99%

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