Definition 1.2.3 (Rectifiable sets) . A set A ⊆ R n is called an m -dimensional rectifiable set if H $^{m}$( A ) < ∞ and there exists a set E such that H $^{m}$( A − E ) = 0 and E is the union of the images of countably many Lipschitz functions from$_{R}$ m to$_{R}$ $^{n}$. Definition 1.2.4 (Density) . Given a set A ⊆ R n and 1 ≤ m ≤ n , the m -dimensional density of A at a point x ∈ R n is given by where B ( x, r ) is the closed ball in R n with center x and radius r and α$_{m}$ is the volume of the unit ball in$_{R}$ $^{m}$. Definition 1.2.5 (Density of measures) . Given a measure µ on R $^{n}$, 1 ≤ m ≤ n , and x ∈ R $^{n}$, we define the m -dimensional measure of µ at x by Density of a set in Definition 1.2.4 is a special case of density of measures using the Hausdorff measure restricted to A (denoted H $^{m}$¬ A and defined by ( H $^{m}$¬ A )( B ) = H $^{m}$( A ∩ B )). The following is a brief introduction to currents, largely following Federer [19], Krantz and Parks [29], and Morgan [32] which are recommended as references for some of the details in descending order of difficulty. Currents are the primary objects of study in Chapters 2 and 3 where the definition of various types of currents (general, normal, and integral) and the flat norm on currents play a central role. There is significant machinery 3 1.3 Currents ϑ $^{m}$( A, x ) = lim r ↓ 0 H $^{m}$( A ∩ B ( x, r )) α$_{m}$r m θ $^{m}$( µ, x ) = lim r ↓ 0 µ ( B ( x, r ) α$_{m}$r m .