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I'm trying to give an example of a bounded set of measure zero that is rectifiable and an example of a bounded set of measure zero that is not rectifiable.

I can't think of 1 the top of my head, and for a susbet S of R^n to be rectifiable, we need to have S bounded and BdS measure zero.

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"I can't think of 1 the top of my head, and for a susbet S of R^n to be rectifiable, we need to have S bounded and BdS measure zero." If all you require is that the set be bounded, have measure zero, and be such that the boundary of the set doesn't have measure zero, than any countable dense set will work. (For a bounded measure zero set that is rectifiable, what about a singleton set?) –  Dave L. Renfro Feb 9 '12 at 19:48
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Uh, why did you flag your own question? Is that a misclick? –  Willie Wong Feb 28 '12 at 9:52
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Rectifiables are somewhat easier to find, so we'll leave them aside for now. As for a non-rectifiable set of measure $0$, you might want to take a look at the set $\mathbb{Q}\cap(0,1)$. This is a concrete and canonical example to the comment of Dave Renfro above.

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