Pointwise versus equal (quasi-normal) convergence via ideals
type: Item
instance of: scholarly article
title: Pointwise versus equal (quasi-normal) convergence via ideals
zbMATH Open document ID: 1314.40007
author: Rafał Filipów
author: Marcin Staniszewski
published in: Journal of Mathematical Analysis and Applications
publication date: 2014-10-31T00:00:00Z
reviewed by: Umit Totur
Mathematics Subject Classification ID: 40A35
Mathematics Subject Classification ID: 40G15
zbMATH DE Number: 6362859
zbMATH Keywords: ideal
zbMATH Keywords: ideal convergence
zbMATH Keywords: equal convergence
zbMATH Keywords: quasi-normal convergence
zbMATH Keywords: pointwise convergence
zbMATH Keywords: bounding number
MaRDI profile type: Publication
full work available at URL: j.jmaa.2014.09.004
OpenAlex ID: W1964568523
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DOI: 10.1016/J.JMAA.2014.09.004
review text: Let \(\mathcal{I},\mathcal{J}\) be proper ideals on \(\mathbb{N}\). Let \(f_n\) \((n\in \mathbb{N})\) and \(f\) be real-valued functions defined on a set \(X\).NEWLINENEWLINEThe sequence \((f_n)\) is \(\mathcal{I}\)-pointwise convergent to \(f\) if \(\{n: |f_n(x)-f(x)|\geq \varepsilon\}\in \mathcal{I}\) for every \(x\in X\) and \(\varepsilon >0\).NEWLINENEWLINEThe sequence \((f_n)\) is \((\mathcal{I},\mathcal{J})\)-equally convergent to \(f\) if there exists a sequence of positive reals \((\varepsilon_n)\) which is \(\mathcal{J}\)-pointwise convergent to \(0\) and \(\{n: |f_n(x)-f(x)|\geq \varepsilon_n\}\in \mathcal{I}\) for every \(x\in X\).NEWLINENEWLINEIn this paper, the authors prove the following theorem:NEWLINENEWLINE{Theorem.} Let \(\mathcal{I},\mathcal{J}\) be ideals on \(\mathbb{N}\). For every set \(X\) the following assertions are equivalent.NEWLINENEWLINE(1) For every sequence \((f_n)\) of real-valued functions defined on \(X\), if \((f_n)\) is \(\mathcal{I}\)-pointwise convergent to \(f\), then \((f_n)\) is \((\mathcal{I},\mathcal{J})\)-equally convergent to \(f\).NEWLINENEWLINE(2) For every family \(\{E_{n}^{\alpha}:n\in \mathbb{N}, \alpha<|X|\}\subseteq \mathcal{I}\) such that \(E_{n}^{\alpha}\cap E_{k}^{\alpha}=\emptyset\) for \(n\neq k\), \(\alpha<|X|\), there exists a partition \(\{A_n: n\in \mathbb{N}\}\subseteq \mathcal{J}\) of \(\mathbb{N}\) such that NEWLINE\[NEWLINE \bigcup_{n\in \mathbb{N}}\left(A_n\cap \bigcup_{i\leq n}E_{i}^{\alpha}\right)\in \mathcal{I} NEWLINE\]NEWLINE for every \(\alpha<|X|\).
description: scientific article; zbMATH DE number 6362859