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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 $q>0$?

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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".

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  • $\begingroup$ Thanks for the links and terminological tip. May I ask, is there typical examples of upper Ahlfors regular measures? $\endgroup$
    – AIM
    Feb 13 at 12:46

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