A characterization of strong measure zero sets
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.
scientific article; zbMATH DE number 878288
scientific article