Covering with sets of negligible boundary

Solution 1:

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!

Thank you!