The hausdorff-outer-measure is defined for all subsets of a metric space. The hausdorff measure is defined as the restriction to caratheodory measurable sets.
I actually don't know how the set of hausdorff measurable sets look like but since n-dimensional hausdorff measure and n-dimensional lebesgue measure coincidence when n is an integer there should be non-measurable sets for hausdorff measure.
However Hausdorff-dimension is often defined for all sets.
What is the Hausdorff-dimension of non measurable sets? Or how is the dimension for such sets even defined?