Terminology "upper" Ahlfors regular measure

Question

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 $q>0$?

Answer 1

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. See, e.g., https://www.maths.ed.ac.uk/~jazzam/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".