Regularities of multifractal measures

First, we prove the decomposition theorem for the regularities of multifractal Hausdorff measure and packing measure in R-d. This decomposition theorem enables us to split a set into regular and irregular parts, so that we can analyze each separately, and recombine them without affecting density properties. Next, we give some properties related to multifractal Hausdorff and packing densities. Finally, we extend the density theorem in [6] to any measurable set.