CHAPTER 3 - Lipschitz Functions and Rectifiable Sets

Abstract

This chapter introduces the m-dimensional surfaces of geometric measure theory called rectifiable sets. These sets have folds, corners, and more general singularities. The relevant functions are not smooth functions as in differential geometry but Lipschitz functions. A function f: Rm → Rn is Lipschitz if there is a constant C such that f(x)-f(y)|≤C|x-y|. The least such constant is called the Lipschitz constant and is denoted by Lipschitz function. The chapter presents the graphs of two typical Lipschitz functions. Rectifiable sets can have countably many rectifiable pieces, perhaps connected by countably many tubes and handles and perhaps with all points in Rn as limit points. From the point of view of measure theory, rectifiable sets behave similar to C1 submanifolds.