**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.