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 $q>0$?



Terminology "upper" Ahlfors regular measure
In the case $\omega(r) = Cr^q$:


* the terminology "upper Ahlfors ($q$-)regular" is certainly in use. See, e.g., <https://arxiv.org/pdf/1803.04819.pdf> .
* the terminology "($q$-)Frostman measure" is also used, referring to [Frostman's lemma](https://en.wikipedia.org/wiki/Frostman_lemma). See, e.g., [https://www.maths.ed.ac.uk/~jazzam/posts/2020-01-14-frostmann/](https://www.maths.ed.ac.uk/%7Ejazzam/posts/2020-01-14-frostmann/) .


In the case of a general $\omega$, I personally do not know of a name. I might call it some kind of "upper mass bound".


