Techniques for Computing Exact Hausdorff Measure with Application to a Sierpinski Sponge in $\mathbb{R}^3$

In this dissertation we aim to perform a detailed study of techniques for the analysis of the exact $s$-dimensional Hausdorff measure of fractal sets and try to provide a reasonably comprehensive review of the required background. An emphasis is placed on results pertaining to local density of sets and we show how these provide a link to the more global concept of Hausdorff measure. A new result is provided which states that if $K$ is a self-similar set satisfying the open set condition, then $\mathcal{H}^s(K \cap U) \leq |U|^s$ for all Borel $U$, also implying that $\overline{D}_c(K, x) \leq 1$ for all $x$, where $\mathcal{H}(E)$ and $\overline{D}_c(E, x)$ refer to the $s$-dimensional Hausdorff measure of some set $E$ and the local convex density of $E$ at a point $x$ respectively. Based on the work of Zuoling Zhou and Min Wu, we provide new calculations for the exact Hausdorff measure of both a Sierpinski carpet in $\mathbb{R}^2$ and a Sierpinski sponge in $\mathbb{R}^3$. In the final chapter we take a look at how the Hausdorff measure behaves when measuring the invariant sets associated with special types of iterated function systems known as iterated function systems with condensation and also provide a brief discussion on the calculation of the packing measure of a self-similar set.