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A characterization of strong measure zero sets. (English) Zbl 0857.28001
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.

28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
Full Text: DOI
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