# Lebesgue-measure for closed subsets of [0,1] Answer

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## Lebesgue-measure for closed subsets of [0,1]

I have to proof that: for each number $$\alpha$$ with $$0<\alpha<1$$ there is a closed subset $$C$$ of $$[0,1]$$ that satisfies $$\lambda(C)=\alpha$$ and includes no non-empty open set.

I think that the construction of the Cantor set might has something to do with it. But I’m not sure where to start.

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