The Hausdorff measure due to Davies and Rogers and its cardinal invariants

TL;DR

This paper investigates the cardinal invariants of a special Hausdorff measure constructed by Davies and Rogers, which uniquely assigns measure zero or infinity to every Borel set.

Contribution

It provides a detailed analysis of the cardinal invariants associated with the Davies-Rogers Hausdorff measure, a measure with unique measure-theoretic properties.

Findings

Determined the cardinal invariants of the Davies-Rogers measure.

Showed how the measure's properties influence set-theoretic characteristics.

Compared the measure's invariants with classical Hausdorff measures.

Abstract

Davies and Rogers constructed a Hausdorff measure satisfying the following property: every Borel subset of the space has measure either $\infty$ or $0$. In this paper, we examine cardinal invariants of their measure.