Geometric measure theory could be described as differential geometry, generalized through measure theory to deal with maps and surfaces that are not necessarily smooth, and applied to the calculus of variations. Geometric measure theory is important because it studies sets, their variation and their boundaries (from the measure theoretic sense). In particular, a very interesting branch in Geometric measure theory, is the sets of locally finite perimeter. Just as their name actually shows, these sets are essentially sets whose perimeter is (locally) finite. In this thesis we start by giving a ...