General Hausdorff functions, and the notion of one-sided measure and dimension

Abstract

The main facts about Hausdorff and packing measures and dimensions of a Borel set E are revisited, using determining set functions $\phi_\alpha\colon\mathcal{B}_E\to(0,\infty)$, where $\mathcal{B}_E$ is the family of all balls centred on E and α is a real parameter. With mild assumptions on φ_α, we verify that the main density results hold, as well as the basic properties of the corresponding box dimension. Given a bounded open set V in ℝ^D, these notions are used to introduce the interior and exterior measures and dimensions of any Borel subset of ∂V. We stress that these dimensions depend on the choice of φ_α. Two determining functions are considered, φ_α(B)=Vol_D(B∩V)diam(B)^α-D and φ_α(B)=Vol_D(B∩V)^α/D, where Vol_D denotes the D-dimensional volume.