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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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    What is a Lebesgue set? You mean Lebesgue-measurable?2012-11-27
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    @MartinArgerami Yes2012-11-27
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    A related question on MathOverflow: http://mathoverflow.net/questions/43721/is-arbitrary-union-of-closed-balls-in-rn-lebesgue-measurable2012-11-27
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    And one of the proof sketches there does use the Vitali covering theorem.2012-11-27

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