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 $\beta=5^{-n}$.
This would imply $\sum_{j=1}^{N}|5Q_j|\geq|E|_e$.
My question is that, does this imply that $\bigcup_{j=1}^{N}5Q_j$ covers $E$?
The picture in Wikipedia seems to suggest so: https://en.m.wikipedia.org/wiki/Vitali_covering_lemma
However I am not able to prove it rigorously. Just because it has greater measure than $E$ doesn't seem to directly imply it is a cover.
Thanks!
(I'm not sure what the notation $|\cdot|_e$ means, but I am pretty sure that no matter what it is, these counterexample should work, possibly with a small tweak.)
This can fail in a very stupid way; let $E=(-1,1)\cup\{400\}$ and $K$ contain only the cubes $[-1,1]$ and $[400,401]$. You can then take $Q_1=[-1,1]$ to satisfy the lemma but fail to get the awkward point.
Such an example could be made less silly by replacing $\{400\}$ with a set of very small measure, say $(400,400.00001)$, and we could then imagine this being a "subscenario" of a more "realistic" situation as well.