In the attatchment below, I defined a new measure between 0 and 1 that measures the uniformity of subset R of [0,1] x [0,1] w.r.t a set that's "uniform" in [0,1] x [0,1] with Hausdorff dimension d between 0 and 2. Note the larger the measure, the lower the non-uniformity of the subset.

Question: After reading sections 0, 1, and 2, I want to know if there is a simpler measure for the uniformity of subsets of [0,1] x [0,1]