Let $(M,d)$ be a metric space (separable, complete, better?) and let $\mu$ be a ($\sigma$-additive, positive, locally finite, regular?) Borel measure on $M$. For $x\in M$ consider the volume growth function $f_x:[0,+\infty)\to[0,+\infty]$ given by $$ f_x(r)=\mu(B(x,r)),\quad B(x,r)=\\\{y\in M\\\,|\quad d(x,y)<r\\\}. $$

Question: What are some standard sufficient conditions on $(M,d,\mu)$ such that $f_x$ is absolutely continuous? I need references.

This will require certain homogeneity of $\mu$ and $(M,d)$, and even then may not be true for all $x\in M$ but only some. Seems like a classical problem but I cannot find it addressed in any of the many treatments of metric spaces. Let me stress again that I am looking for a reference where this problem is considered.

Thank you.

Let $(M,d)$ be a metric space (separable, complete, better?) and let $\mu$ be a ($\sigma$-additive, positive, locally finite, regular?) Borel measure on $M$. For $x\in M$ consider the volume growth function $f_x:[0,+\infty)\to[0,+\infty]$ given by $$ f_x(r)=\mu(B(x,r)),\quad B(x,r)=\{y\in M\,|\quad d(x,y)<r\}. $$

Question: What are some standard sufficient conditions on $(M,d,\mu)$ such that $f_x$ is absolutely continuous? I need references.

This will require certain homogeneity of $\mu$ and $(M,d)$, and even then may not be true for all $x\in M$ but only some. Seems like a classical problem but I cannot find it addressed in any of the many treatments of metric spaces. Let me stress again that I am looking for a reference where this problem is considered.

Thank you.