TITLE: Does the boundary of metric balls have measure $0$ in a metric measure space with Radon measure?
QUESTION [5 upvotes]: Let $(X,d,m)$ be a metric measure space such that $m$ is a Radon measure and let $\overline{B}(x,r):=\{y\in X \mid d(x,y)\leq r\}$. Is it true that the "boundary" of $\overline{B}$ (meaning the set $\{y\in X \mid d(x,y)=r\}$) has $m$ measure $0$?

REPLY [2 votes]: No. For a trivial example consider a finite metric space.
For a less trivial example, consider $\Bbb R$ with Lebesgue measure, but
with the metric $d(x,y)=\min(|x-y|,1)$, which generates the usual topology.