Dimension Function - Motivation: s-dimensional Hausdorff Measure

Motivation: s-dimensional Hausdorff Measure

Consider a metric space (X, d) and a subset E of X. Given a number s ≥ 0, the s-dimensional Hausdorff measure of E, denoted μs(E), is defined by

where

μ_δs(E) can be thought of as an approximation to the "true" s-dimensional area/volume of E given by calculating the minimal s-dimensional area/volume of a covering of E by sets of diameter at most δ.

As a function of increasing s, μs(E) is non-increasing. In fact, for all values of s, except possibly one, Hs(E) is either 0 or +∞; this exceptional value is called the Hausdorff dimension of E, here denoted dim_H(E). Intuitively speaking, μs(E) = +∞ for s < dim_H(E) for the same reason as the 1-dimensional linear length of a 2-dimensional disc in the Euclidean plane is +∞; likewise, μs(E) = 0 for s > dim_H(E) for the same reason as the 3-dimensional volume of a disc in the Euclidean plane is zero.

The idea of a dimension function is to use different functions of diameter than just diam(C)s for some s, and to look for the same property of the Hausdorff measure being finite and non-zero.