I am studying causality theory in Lorentzian length spaces, and I have a question about geometric measure theory in general (it will help me with a proof I am trying to finish):

Suppose we have a Polish space $(X,d)$, two compactly supported probability measures $\mu,\nu\in\mathscr{P}(X)$ and an optimal coupling $\boldsymbol{\pi}\in\mathscr{P}(X\times X)$ between them, i.e. the marginals of $\boldsymbol{\pi}$ are precisely $\mu$ and $\nu$ and it solves the problem$$\inf\bigg\{\sqrt{\int_{X\times X}d(x,y)^2\,\mathrm{d}\boldsymbol{\pi}(x,y)},\quad\boldsymbol{\pi}\text{ is a coupling between $\mu$ and $\nu$}\bigg\},$$which is the usual $2$-Wasserstein distance.

In my context $S:=\text{supp }\mu\times\text{supp }\nu\subset A$ where $A$ is a particular open subset of $X$. What I wanted to do is to take a finite covering of $S$ with rectangles of the form $B_\varepsilon(x_i)\times B_\varepsilon(y_i)$ where the $B_\varepsilon(z)$ are balls centered at $z\in X$ of radius $\varepsilon>0$.

Now the question: assume to have a locally finite Radon measure $\mathfrak{m}$ on $X$. Is it then possible to consider a covering of $\text{supp }\boldsymbol{\pi}$, say $\{B_i\}$, with pairwise disjoint $B_i$, each $B_i\in B_\varepsilon(x_i)\times B_\varepsilon(y_i)$ and such that $\boldsymbol{\mathfrak{m}(\partial B_i)=0}$?

My doubt is about the $\mathfrak{m}$-measure of the boundaries, altough in my context it is valid the Bishop-Gromov volume estimate (see Sturm's "On the geometry of metric measure spaces II", Proposition 2.3 a reference).

## Best Answer

For anyone that want some answer from that, look at (as Literally an Orange says) Peter Walters' "Introduction to Ergodic Theory", Lemma $8.5$ (page $188$). I copy-paste the statement:

Lemma 8.5: Let $X$ be a compact metric space and $\mu\in\mathscr{P}(X)$. ThenI) if $x\in X$ and $\delta>0$ there exists a $\delta'<\delta$ such that $\mu\big(\partial B(x,\delta')\big)=0$;II) if $\delta>0$ there is a finite partition $\mathcal{A}=\{A_1,\dots,A_k\}$ of $X$ such that $\text{diam }(A_j)<\delta$ and $\mu(\partial A_j)=0$ for each $j=1,\dots,k$.Since our set is compact and $\mathfrak{m}$ is locally finite we can restrict it to the $\text{supp }\boldsymbol{\pi}$ and renormalize it to get a probability measure and apply the Lemma. There's more information in the section $8.2$ of the book that I need to read but this can help a lot!

Thank you!