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ON MINKOWSKI MEASURABILITY
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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 ℝ2 of Minkowski dimension d and arbitrary positive Minkowski content. We also indicate how this construction can be extended to ℝn, 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.
Title: ON MINKOWSKI MEASURABILITY
Description:
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 ℝ2 of Minkowski dimension d and arbitrary positive Minkowski content.
We also indicate how this construction can be extended to ℝn, 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.
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