ON minkowski measurability

Two "pathological" properties of Minkowski content are that countable sets can have positive content (unlike Hausdorff measures) and the property of a set being Minkowski measurable is quite rare. In this paper, we explore both of these issues. In particular, for each d ∈ (0,2) we give an explicit construction of a countable Minkowski measurable subset of ² of Minkowski dimension d and arbitrary positive Minkowski content. We also indicate how this construction can be extended to ⁿ, to construct a countable subset with arbitrary positive Minkowski content of any dimension in (0, n). Furthermore, we give an example of a strictly increasing C¹ function which takes a Minkowski measurable subset of [0,1] onto a set which is not Minkowski measurable but of the same dimension.