Wheedon's Analysis book has this lemma called "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 $\sum_{j=1}^{N}|Q_j|\geq\beta|E|_e$, where $0<\beta<5^{-n}$.
My question is: How does this "Simple Vitali Lemma" relate to the more commonly seen Vitali Covering Lemma e.g. https://en.m.wikipedia.org/wiki/Vitali_covering_lemma?
Is it stronger/weaker/equivalent to Vitali Covering Lemma? It is clear to me that they are similar in spirit, but I am not sure to what extent.
Thanks. A quick google search seems that "Simple Vitali Lemma" is quite rarely found in books, as compared to "Vitali Covering Lemma".