A criterion for N- rectifiable sets

Abstract

Geometric measure theory was developed in the second half of the 20th century to manipulate the structure and regularity questions in the calculus of variations. The main goal of this thesis is to introduce the theory of " rectifiability of sets". Rectifiable sets are considered smooth in a certain measure theoretic sense. Rectifiable sets are basic concepts in geometric measure theory. Their theory began with the study and determination of length, area or volume of sets in Euclidean space. Rectifiable sets have many of the desirable properties that smooth sets have. In this thesis, we will discuss one of their most important features which is the existence of what we call approximate tangent planes. In fact, we will show that a set that has an n-dimensional approximate tangent plane at almost every point is n-rectifiable.