Abstract: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 $f(X)$.
Keywords: meager-additive; $\mathcal E$-additive; strong measure zero; sharp measure zero; Hausdorff dimension; Hausdorff measure
DOI: DOI 10.14712/1213-7243.2015.277
AMS Subject Classification: 03E05 03E20 28A78