# Covering with sets of negligible boundary

geometric-measure-theorymeasure-theorymetric-spacespolish-spaceswasserstein

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).

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 $$\mu\in\mathscr{P}(X)$$. Then
• I) 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!