A criterion for sets of locally finite perimeter

Abstract

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 formal definition for sets of locally finite perimeter. Moreover, we will use the Hausdorff measure ( just like the surface measure ) as a tool to give us the perimeter of the (measure theoretic) boundaries of these sets. Then we will prove a criteria for sets of locally finite perimeter, which states that a set is of locally finite perimeter, if and only if, (locally) the Hausdorff measure of its ( measure theoretic ) boundary is finite.