ABSTRACT
This chapter describes the concepts of length, area, volume, as well as higher dimensional analogues and abstractions of the ideas quite systematically. It discovers relations between objects in the form of theorems. The chapter provides a toolbox of tremendous potential. It characterises the class of Borel sets in a more explicit fashion and for general topological spaces, as the smallest "appropriate" class of sets which contains the family of open sets. The chapter defines for any set in ℝd its area, volume, or a higher dimensional analogue of these notions, starting from rectangles and cuboids. In particular, herein the chapter considers the fundamental measure on ℝd associated to the volume which is called the classical Lebesgue measure on ℝd.
