Unions of disjoint open sets.

Question

Let $X$ be a compact metric space (hence separable) and $\mu$ a Borel probability measure.

Given an open set $A$ and $r,\epsilon>0$ $\ $does there exist a finite set of disjoint open balls $\left\{ B_{i}\right\} $ contained in $A$ and of radius smaller than $r$ , so that $\mu(\cup B_{i})\geq\mu(A)-\epsilon.$

Answer 1

Recall

Lemma (Finite Vitali covering lemma) Let $(X,d)$ be a metric space, $\{B(a_j,r_j),j\in [K]\}$ a finite collection of open balls. We can find a subset $J$ of $[K]$ such that the balls $B(a_j,r_j),j\in J$ are disjoint and $$\bigcup_{i\in [K]}B(a_i,r_i)\subset \bigcup_{j\in J}B(a_j,3r_j).$$

A proof is given page 41 in the book Ergodic Theory: with a view towards Number Theory, Einsiedler M., Ward T.

For each $a\in A$, fix $r_a<r/3$ such that $B(a,3r_a)\subset A$. As $X$ is separable, we can, by Lindelöf property, extract from the cover $\{B(a,r_a),a\in A\}$ of $A$ a countable subcover $\{B(a_j,r_j),j\in \Bbb N\}$. Now take $N$ such that $\mu(A)-\mu\left(\bigcup_{j=0}^NB(a_j,r_j)\right)<\varepsilon$. Then we conclude by finite Vitali covering lemma.