Hausdorff measures provide an important source of examples of Radon measures. For this reason, before further developing the theory of Radon measures, in this chapter we take a closer look at Hausdorff measures. In Section 3.1 we introduce and motivate the notion of Hausdorff dimension. In Section 3.2 we relate 1 to the classical notion of length of a curve. Finally, in Section 3.3 we show equivalence of the Lebesgue measure and n on n.
3.1 Hausdorff measures and the notion of dimension
We begin our discussion by introducing ...