# Does every compact doubling metric space have a canonical measure?

My question is this one, with the additional condition that the metric space be doubling. In the aforementioned question, the limiting measure depends on the sequence $$\epsilon_n$$ and hence is not canonical.

The metric space $$(X,\rho)$$ is "doubling" (with dimension $$d$$) if every radius $$r$$-ball in $$X$$ can be covered by $$2^d$$ radius $$r/2$$-balls.

In the question you linked to, the answer by user "R W" provides an example of a compact metric space for which the requested construction provides different measures for different sequences $$(\epsilon_n)$$.
R W's construction is the boundary (set of geodesic rays) of a tree with maximal valence $$4$$, where the distance $$d(x,x')$$ is $$2^{-d}$$ for $$d$$ the number of edges in the intersection of the two rays.
It's not hard to see that R W's space is doubling. A ball of radius $$r=2^{-d}$$ in $$X$$ consists of all geodesic rays sharing an initial segment of length $$d$$ with a given ray. There are only at most $$4$$ possible initial segments of length $$d+1$$ among those rays, so the ball of radius $$r$$ is covered by at most four balls of radius $$r/2$$.
For $$r$$ not a power of $$2$$ the result then follows easily.