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Let E(n) be n-dimensional Euclidean space. It is known that there exist subsets of E(n) which are simple arcs and have positive n-dimensional Lebesgue measure when n=1 or 2. Does this continue to be true for arbitrarily large n? If not, what is the largest n for which it holds and is there a simple proof of this fact? Intuitively, I feel that there should be no upper bound, but cannot see how to prove it.

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Yes (this is so for every $n=1\ 2\ \ldots$). – Włodzimierz Holsztyński May 7 '13 at 0:04
Thanks alot for your answer. I have searched extensively to find it in the literature but with no success. – Garabed Gulbenkian May 7 '13 at 19:42

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