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Show that the n-dimensional Hausdorff measure of an $n$-dimensional cube is positive and finite. I can easily show that if it is finite then the $n+1$ dimensional measure is $0$ and the $n-1$ dimensional measure is $\infty$, but I'm not sure how to show that it is at exactly $n$ that the positive finite case occurs. Can anyone provide any tips?

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    The harder direction is "positive". For that you can use the existence of Lebesgue measure to get a lower bound.2012-11-28

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More detail for my comments from 2 years ago. Since the OP merely asked for "tips" I didn't provide a complete proof then.

finite
Take an $n$-dimensional cube $C$ of side $a>0$. Fix $\epsilon>0$. Let $k$ be a positive integer. Then $C$ is the union of $k^n$ cubes of side $a/k$. Each of those small cubes has diameter $a\sqrt{n}/k$. Choose $k$ large enough that this is $< \epsilon$. So, using this cover of $C$, we get $ \mathcal{H}^n_\epsilon(C) \le k^n \left(\frac{a\sqrt{n}}{k}\right)^n = a^n n^{n/2}. $ This is true for all $\epsilon>0$, so $\mathcal{H}^n(C) \le a^n n^{n/2}$. Finite.

positive
Write $m$ for the $n$-dimensional Lebesgue outer measure. Note: If $A$ is a set of diameter $t$, then $A$ is contained in a closed ball of radius $t$. (Namely any such ball whose center is a point of $A$.) Let $k$ be the measure of a ball of radius $1$. So if $A$ has diameter $t$, then $m(A) \le kt^n = k({\rm diam }\;A)^n$. Now let $C$ be a cube with side $a$. Consider a countable cover $C \subseteq \bigcup_{i=1}^\infty A_i$. Then $ a^n = m(C) \le \sum_{i=1}^\infty m(A_i) \le k\sum_{i=1}^\infty ({\rm diam}\;A_i)^n $ so $ \sum_{i=1}^\infty ({\rm diam}\;A_i)^n \ge \frac{a^n}{k} . $ This is true for all countable covers of $C$, so $\mathcal{H}^n(C) \ge a^n/k$. Positive.

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    This was very enlightening. For some reason the relation between the Lebesgue measure and $\operatorname{diam}$ escaped me. Thank you. (I have to wait to award bounty).2014-11-20
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you can use the relationship between Hausdorff n-dimensional measure and Lebesgue Measure . http://en.wikipedia.org/wiki/Hausdorff_measure