In this thesis, we will prove a very important theorem in real analysis called The Area formula for the Hausdorff Measure. This theorem is an extension of the well known theorem the Change of variables formula for the Lebesgue measure. In this thesis, we will define the Hausdorff measure and prove some of its properties. We will also define Lipschitz functions and prove some of its properties also. Then, we continue the thesis by proving all the lemmas needed to finalize the proof of the Area formula for the Hausdorff measure. Finally, we finish this thesis by showing three applications of the ...

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 ...