ABSTRACT
This selection contains a discussion of the relation between Hausdorff dimension and existence of finite measures with metric properties. If there is a finite nonzero measure φ concentrated on the compact set https://www.w3.org/1998/Math/MathML"> E ⊆ ℝ q https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429037252/8de01632-0afb-4bad-a956-f9a9ec6de7c2/content/ieq0501.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , and there are positive constants k and p such that, for every q-dimensional cube R, the inequality φ(R) ≤ k(diameter R)p holds, then the Hausdorff dimension of E is ≥ p.
