1.2 Measure theory A few concepts from measure theory prove useful in our development. The Hausdorff measure allows us to sensibly measure m -dimensional sets in R $^{n}$. Definition 1.2.1 (Hausdorff measure) . Given a set A ⊂ R $^{n}$, the m -dimensional Haus- dorff measure of A is an outer measure defined by where α$_{m}$ is the volume of the unit ball in$_{R}$ m and the infimum is taken over all countable coverings S = { S$_{1}$,S$_{2}$,... } of A with every S$_{j}$ ∈ S having diameter at most δ . The Hausdorff measure approximates A locally by covering it with small sets which in turn have their m -dimensional volumes approximated by balls of the same radius in R $^{m}$. This is the natural way to measure m -dimensional volume in R n and agrees with intuitive notions of what this should mean, for example, for an m -dimensional manifold embedded in R $^{n}$. It also provides sensible results for any nonnegative real dimension by extending the unit ball volume via the Γ function: α$_{m}$ = π m/ $^{2}$/ Γ( m/ 2 + 1). For any particular nonempty set A , there is a “correct” dimension to use when measuring it with the Hausdorff measure in the sense that using any other value yields a trivial result. Definition 1.2.2 (Hausdorff dimension) . The Hausdorff dimension of a nonempty set A ⊆ R n is the unique nonnegative real number m such that H $^{p}$( A ) = 0 for all p > m and H $^{q}$( A ) = ∞ whenever m > q and q ≥ 0 . Knowing that the set A has Hausdorff dimension m places no restrictions on H $^{m}$( A ). That is, one can construct examples with any desired measure in the interval [0 , ∞ ]. 2 H $^{m}$( A ) = lim δ ↓ 0  $_{}$inf S ∑ S$_{j}$ ∈S α$_{m}$$^{(}$ diameter S$_{j}$ 2 ) m  