The Vitali Covering Theorem in $\mathbb{R}^n$

We work in the setting of Euclidean Space and Lebesgue Measure: Standing Notation, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the Euclidean norm $\lVert\,\cdot\,\rVert$, distance $d_{E}$, topology, notion of openness and closed balls $\bar{B}(y,r)$ on $\mathbb{R}^{n}$, the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{n})$, and Lebesgue measure $\lambda_{n}$ are as fixed there. Write $\lambda_{n}^{\ast}$ for Lebesgue outer measure.

Let $E\subseteq\mathbb{R}^{n}$ satisfy $\lambda_{n}^{\ast}(E)<\infty$, and let $\mathcal{F}$ be a set of pairs $(y,r)$ with $y\in\mathbb{R}^{n}$ and $r\in\mathbb{R}$, $0<r$; the pair $(y,r)$ is said to carry the closed ball $\bar{B}(y,r)$. Suppose $\mathcal{F}$ finely covers $E$, meaning ¶ that for every $x\in E$ and every $\eta\in\mathbb{R}$ with $0<\eta$ there is $(y,r)\in\mathcal{F}$ with

Then the following holds. ¶ For every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there are a natural number $N$ and pairs $(y_{1},r_{1}),\dots,(y_{N},r_{N})\in\mathcal{F}$ whose closed balls are pairwise disjoint and satisfy

where $N$ may be $0$, the union over an empty range being empty.