  We develop a theory of \emph{sharp measure zero} sets that parallels Borel's
\emph{strong measure zero}, and prove a theorem analogous to
Galvin-Myscielski-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
$f:2^\omega\to2^\omega$ is continuous and $X$ is meager-additive, then so is
$f(X)$.
