Simple Vitali Lemma

Let $E$ be a subset of $\mathbb{R}^n$ with $|E|_e<\infty$ , and let $K$ be a collection of cubes $Q$ covering $E$ . Then there exist a positive constant $\beta$ (depending only on $n$ ), and a finite number of disjoint cubes $Q_1,\dots,Q_N$ in $K$ such that $\displaystyle \sum_{j=1}^N|Q_j|\geq\beta|E|_e.$
(We may take $0<\beta<5^{-n}$ .)