Let $\mathcal{K}$ be ,not necessarily countable, a family of compact cubes in $\mathbb{R}^N$. How to show that $\bigcup${$K:K\in\mathcal{K}$} is a Lebesgue measurable set?
Here all cubes are nondegenerate.
I think it may be necessary to use the Vitali's covering Theorem. But I am not sure how to use it. Can someone give some hints?