  We show that if the upper Assouad dimension of the compact set $E\subseteq
\mathbb{R}$ is positive, then given any $D>\dim_{A}E$ there is a measure with
support $E$ and upper Assouad (or regularity) dimension $D$. Similarly, given
any $0\leq d<\dim_{L}E,$ there is a measure on $E$ with lower Assouad dimension
$d$.
