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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?

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    And one of the proof sketches there does use the Vitali covering theorem.2012-11-27

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Do these have nonvoid interiors? If not, you can form an arbitrary subset of $\mathbb{R}^N$ with such a union.

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    @JonasMeyer Here all cubes are nondegenerate. All cubes have positive lebesgue measure.2012-11-27