Lebesgue Measure in r-Space

Abstract

By an interval I in Euclidean r-space (r = l, 2, …) is meant a rectangular parallelepiped with edges parallel to the axes. It is the Cartesian product of r 1-dimensional intervals. As in the 1-dimensional case, the r-dimensional volume of I will be denoted by |I|. Lebesgue measure in r-space is an extension of the notion of volume to a larger class of sets. Thus Lebesgue measure has a different meaning in spaces of different dimension. However, since we shall usually regard the dimension as fixed, there is no need to indicate r explicitly in our notations.