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Strong measure zero and meager-additive sets through the prism of fractal measures. (English) Zbl 07088828
Summary: We develop a theory of sharp measure zero sets that parallels Borel’s strong measure zero, and prove a theorem analogous to Galvin-Mycielski-Solovay theorem, namely that a set of reals has sharp measure zero if and only if it is meager-additive. Some consequences: A subset of $$2^{\omega}$$ is meager-additive if and only if it is $$\mathcal{E}$$-additive; if $$f\colon 2^{\omega}\to 2^{\omega}$$ is continuous and $$X$$ is meager-additive, then so is $$f(X)$$.

##### MSC:
 03E05 Other combinatorial set theory 03E20 Other classical set theory (including functions, relations, and set algebra) 28A78 Hausdorff and packing measures
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