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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.

##### MSC:
 28A05 Classes of sets (Borel fields, $$\sigma$$-rings, etc.), measurable sets, Suslin sets, analytic sets
##### Keywords:
measure zero sets; meager sets; strongly null sets
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##### References:
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