A characterization of strong measure zero sets
type: Item
instance of: scholarly article
title: A characterization of strong measure zero sets
zbMATH Open document ID: 0857.28001
DOI: 10.1007/BF02761100
author: Janusz Pawlikowski
published in: Israel Journal of Mathematics
publication date: 1997-02-26T00:00:00Z
review text: A subset \(X\) of the real line \(\mathbb{R}\) is strongly null (has strong measure zero) if for every sequence \(\varepsilon_1,\varepsilon_2,\dots,\varepsilon_n,\dots\) of positive real numbers, there exists a sequence \(X_1,X_2,\dots,X_n,\dots\) of subsets of \(\mathbb{R}\) such that \(X\subseteq \bigcup_n X_n\) and \(\text{diam}(X_n)<\varepsilon_n\). In this paper, the author gets the following new characterization of strongly null sets: Theorem. A set \(X\) in \(\mathbb{R}\) is strongly null if and only if \(F+X\) is null for every closed null set \(F\) in \(\mathbb{R}\). The last section of this paper is devoted to several interesting notes and an open question.
Mathematics Subject Classification ID: 28A05
zbMATH DE Number: 878288
zbMATH Keywords: measure zero sets
zbMATH Keywords: meager sets
zbMATH Keywords: strongly null sets
reviewed by: Barthélemy Le Gac
MaRDI profile type: Publication
cites work: Q4299995
cites work: On covering of real line by null sets
cites work: Closed measure zero sets
cites work: Strong Measure Zero and Strongly Meager Sets
cites work: On ideals of subsets of the plane and on Cohen reals
cites work: Q3715868
cites work: Q3842470
cites work: Every null-additive set is meager-additive
description: scientific article; zbMATH DE number 878288