CHAPTER 2 - Measures

Abstract

This chapter discusses the measure-theoretic foundation and presents the definition of Hausdorff measure and covering theory. There is a unique Borel regular, translation invariant measure on Rn such that the measure of the unit cube [0, l]n is 1. This measure is called Lebesgue measure. In 1918, F. Hausdorff introduced an m-dimensional measure in Rn that gives the same area for submanifolds but is defined on all subsets of Rn. When m = n, it turns out to be equal Lebesgue measure. In 1932, J. Favard defined another m-dimensional measure on Rn (m = 0, 1, … , n), which is called integral-geometric measure. It turns out that integral-geometric measure agrees with Hausdorff measure on all smooth m-dimensional submanifolds and other nice sets but disagrees and often is zero on Cantor-like sets.