Uma introdução à teoria geométrica da medida: fórmulas da área e coárea e verificabilidade

Abstract

The objective of this work is to introduce Geometric Measure Theory (GMT), with a focus on the Area and Coarea formulas. The area formula can be viewed as a generalization of the change of variables theorem for multiple integrals, typically studied in advanced Analysis courses. The coarea formula, in turn, is a generalization of the Fubini-Tonelli theorem covered in Measure Theory courses, and includes the area formula as a particular case. These results deal, a priori, with Lipschitz maps, but can also be stated for maps of class C 1 . The first chapter is dedicated to reviewing concepts and surveying results used throughout the text. Since these are typically covered in an introductory measure theory course, most will be presented without proofs. In the second chapter, we lay the groundwork to present the main results. Finally, in the third chapter, we discuss rectifiable sets in order to state and prove the area and coarea formulas within this even more general context.