Multifractal dimension inequalities in a probability space

Let ν be a probability measure on Ω. We define the upper and lower multifractal box dimension (the measure ν with respect to μ) on a probability space and investigate the relation between the multifractal box dimension and the multifractal Hausdorff dimension, the multifractal pre-packing dimension . Then, we generalize the dimension inequalities of multifractal Hausdorff measures and multifractal packing measures in a probability space.