## Thoughts on Hausdorff Measure and Hausdorff Dimension

The Hausdorff measure is a good way to study the fine properties of sets since it is more ‘sensible’ than the Lebesgue measure. For example in the surfaces and paths have zero Lebesgue measure, but the Hausdorff dimension captures what Lebesgue measure misses for zero-measure sets, and it confirms our intuition that a path is in fact one dimensional and a surface is two dimensional. The Hausdorff dimension of a set is the exact value of such that the -dimensional Hausdorff measure is zero for and infinity for . The problem of finding the Hausdorff dimension of a set is not very simple, since the dimension doesn’t need to be an integer. There exist sets with fractional and other bizare dimensions. I will present a few problems below which can be approached using Hausdorff measure and Hausdorff dimension. A good reference for starters can be found in: Real Analysis.

**1. **Prove that if is -Holder continuous surjective mapping then .

*Proof: *The proof of this problem relies on a simple lemma, which says that Holder continuous functions and Hausdorff measures are strongly connected.

**Lemma: **Suppose that defined on a compact set of some is -Holder continuous. Then

- if .
- .

*Proof of the lemma: *Suppose is a countable family of sets that covers and $m_\alpha(E)$ is finite. Then covers and moreover has diameter less than . Therefore

and (i) now follows. The second part is merely a consequence of the first part.

Now the solution of the problem is a direct consequence of point (ii) in the lemma. Indeed, since is -Holder continuous, we have .

**2. **Prove that there is no mapping such that for all .

*Proof: *Suppose such a mapping does exist. Then note that it is injective, i.e. one to one. Denote by the image of in . Then there exists the inverse mapping which is onto and satisfies . Apply again the second part of the lemma for , where is a sequence of compacts which cover . Then we get , where we have used the fact that the Hausdorff dimension of a countable union of sets is less or equal than the supremum of the Hausdorff dimension taken on the considered sets. We have reached a contradiction, and therefore there isn’t such a function.