Intermediate value property for the Assouad dimension of measures

Hare, Mendivil, and Zuberman have recently shown that if $X$ is a compact subset of the reals and of non-zero Assouad dimension ${\dim }_{A}X$, then for all $s>{\dim }_{A}X$, $X$ supports measures with Assouad dimension $s$. We generalise this result to arbitrary complete metric spaces.