Comparing the Uniformity Invariants of Null Sets for Different Measures

The uniformity invariant for Lebesgue measure is defined to be the least cardinal of a non-measurable set of reals, or, equivalently, the least cardinal of a set of reals which is not Lebesgue null. This has been studied intensively for the past 30 years and much of what is known can be found in [?] and other standard sources. Among the well known results about this cardinal invariant of the continuum is that it can equally well be defined using Lebesgue measure on R without changing the value of the cardinal. Indeed, equivalent definitions will result by using any Borel probability measure on any Polish space. However, the question of the values of uniformity invariants for other, non-σ-finite Borel measures is not so easily answered. This paper will deal with the most familiar class of such measures, the Hausdorff measures for fractional dimension. Observe that by the previous remarks, the least cardinal of any non-measurable subset of any σ-finite set will be the same as the uniformity invariant for Lebesgue measure. In other words, this paper will be concerned with the uniformity invariant of the ideal of σ-finite sets with respect to a Hausdorff measure.