CHAPTER 4 - Normal and Rectifiable Currents

Abstract

To define boundary and establish compactness properties, it will be useful to view rectifiable sets as currents, that is, linear functionals on smooth differential forms (named by analogy with electrical currents). Currents associated with certain rectifiable sets, with integer multiplicities, will be called rectifiable currents. The mass of a rectifiable current is just the Hausdorff measure of the associated rectifiable set (counting multiplicities). The larger class of normal currents will allow for real multiplicities and smoothing. This chapter provides an overview of normal and rectifiable currents. It presents a table that gives spaces of currents that play an important role in geometric measure theory.