ABSTRACT
During the golden era of measure theory (namely the first two decades of the 20th century), Carathe´odory was the first to consider the notion of “length” for sets in RN . Later, in 1919, Hausdorff, motivated by the ideas of Carathe´odory, introduced the measure and dimensional concepts that we shall discuss in this chapter. So in the modern language, the “length” of a set A ⊆ RN will be its Hausdorff one-dimensional outer measure (denoted by µ(1)). Following the pioneering works of Carathe´odory and Hausdorff, significant contributions to the subject were made by Besicovitch. In fact, in the first decade of development of the subject, the main advances on the subject were made by Besicovitch and his students, since geometric measure theory was not part of the mainstream measure theory. However, since the early 70’s, the subject attracted a large number of researchers, due to its fundamental importance in the study of the so-called “Fractal Geometry.” Fractal sets arise in many applications, such as turbulence in fluids, geographical coastlines and surfaces, fluctuation of prices in stock exchanges, the Brownian motion of particles and others. Mandelbrojt was the first to emphasize their use to model a variety of phenomenona. There have been many ways to estimate the “size” or “dimension” of small (thin) sets and of highly irregular sets and to generalize the idea that points, curves and surfaces have dimensions 0, 1 and 2 respectively. Hausdorff measure has the advantage of being a measure and together with the notion of Hausdorff dimension can provide a more delicate sense of the size of sets in RN than Lebesgue measure provides. To illustrate this, consider in R2 the set
A df = {(
t, sin 1 t
) : t ∈ (0, 1)
} .
