Let $(X,d)$ be a metric space and $m$ be a Borel measure on $(X,d)$. The measure $m$ is called Ahlors regular if $m(B(x,r))\asymp r^q$ for some $q>0$ and each $x\in X$. Is there a name for measures satisfying the following relaxation:

There is some strictly monotone continuous increasing function $\omega:[0,\infty]\rightarrow [0,\infty]$ with $\omega(0)=0$ and $\omega(\infty)=\infty$ satisfying: $$ m(B(x,r)) \leq \omega(r); $$ what about in the simple case where $\omega(r)=r^q$ for some $q>0$?