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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.

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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.

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