# Terminology "upper" Ahlfors regular measure

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 $$q>0$$?

In the case $$\omega(r) = Cr^q$$:
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".