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Edit: Shortly after this post it was rightly pointed out by @AntonPetrunin that the measure $\mu$ may not be unique. @R W then showed how one can construct a metric space where the limiting measure is not unique. Hence @R W answered the following question in the negative: Does the following construction always produce a unique limiting measure? Therefore the construct below doesn't produce a canonical probability measure for every compact metric space (as was claimed below). Thank you @AntonPetrunin and @R W for catching the error and providing an example.

Original Post:

My coauthor Sean Li and I recently ran across the (seemingly not well-known) fact that every compact metric space is endowed with a canonical probability measure. The construction has been used to prove the existence of Haar measures on compact groups (see for instance page 3 of Assaf Naor's notes). However, we're not able to find a reference for a general compact metric space. Does anyone have a reference?

Let $(X,d)$ be a compact metric space, let $\epsilon>0$, and let $Y_{\epsilon}\subset X$ be an $\epsilon$-net of minimal cardinality. That is, $\displaystyle\cup_{y\in Y_{\epsilon}}B(y,\epsilon)=X$ where $B(y,\epsilon)$ is the ball of radius $\epsilon$ centered at $y$, and $\# Y_{\epsilon}$ is minimal with respect to all such sets.

Let $\mu_{\epsilon}$ be the Borel probability measure on $X$ defined by $\mu_{\epsilon}(E)=(\# E\cap Y_{\epsilon})/\# Y_{\epsilon}$. By compactness it follows that there exists a Borel probability measure $\mu$ on $X$ such that $\mu$ is the weak limit (up to taking a subsequence) of $\mu_{\epsilon}$ as $\epsilon\to 0$.

Theorem The measure $\mu$ is unique. That is, it doesn't depend on the choice of minimal $\epsilon$-nets $Y_{\epsilon}$.

A proof is supplied in the notes of Naor cited above on page 4 and 5, during the course of his proof of existence of Haar measure. According to Naor, he got this proof from Milman--Schechtman (page 1 and 2), and Milman and Schechtman cite the paper "Abstrakte fastperiodische Funktionen" of Maak (Abh. Math. Sem. -- 1936). Naor seems to be the first to point out that such a unique measure didn't rely on the underlying homogeneous structure of the metric space (see the comments in his notes). Can anyone point us to a reference (besides the notes of Naor) of this fact? Also, we're not able to track down a copy of Maak's paper to check if there is any mention of any of this there.

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    $\begingroup$ The construction depends on the choice of the sequence $\varepsilon_n\to 0$, is not it? So you can not say that the measure is canonical. $\endgroup$ – Anton Petrunin Aug 9 '17 at 19:58
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    $\begingroup$ @AntonPetrunin Yes - it seems so. The only relevant part of Naor's exposition is the fact that any two minimal $\epsilon$-nets are $2\epsilon$-close. $\endgroup$ – R W Aug 9 '17 at 20:44
  • $\begingroup$ @AntonPetrunin, you're completely right. Thanks for pointing out my error. @R W thank you for providing an example showing that there may be several limit points. I'll add a note at the end of my post asking the question and accepting your answer. $\endgroup$ – M. Kelly Aug 10 '17 at 1:31
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As Anton has already mentioned, one can only claim that if the sequence $\mu_{\epsilon_n}$ associated to a certain sequence of minimal $\epsilon_n$-nets $Y_n$ converges, then it will also converge to the same limit for for any other sequence of $\epsilon_n$-nets. However, a different sequence $(\epsilon_n)$ may produce a different limit measure.

For the simplest counterexample let $T$ be the (genealogical) tree constructed in the following way: the progenitor $o$ has two first generation descendants $a,b$. Further, in the branch starting from $a$ (resp., $b$) everyone in even generations (counted with respect to $o$) has 4 (resp., 2) descendants, and everyone in odd generations has 2 (resp., 4) descendants. Let now $X=\partial T$ be the boundary of this tree ($\equiv$ the set of infinite geodesic rays issued from $o$ $\equiv$ the set of all inifnite lines of descendants starting from $o$) endowed with the metric $$ d(x,x') = 2^{-(x|x')} \;, $$ where $(x|x')$ denotes the confluence of the rays $x$ and $x'$ (i.e., the length of their common part). Then OP's construction produces two limit measures corresponding to the sequences $\epsilon_n=2^{-2n}$ and $\epsilon_n=2^{-2n+1}$. In this case the two limit measures are equivalent, but one can easily obtain examples with singular limit measures, with inifnitely many limit measures, etc, etc.

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