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

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. Our main results are that the covering number of this Hausdorff measure zero ideal is below the uniformity of the ideal of closed Lebesgue measure zero ideal and its dual. Also we showed the strict inequality between them is consistent.