Different types of multifractal measures in separable metric spaces and their applications

The properties of various fractal and multifractal measures and dimensions have been under extensive study in the real-line and higher-dimensional Euclidean spaces. In non-Euclidean spaces, it is often impossible to construct non-trivial self-similar or self-conformal sets, etc. We consider in the present paper the proper way to phrase the definitions for use in general metric spaces. We investigate the relative Hausdorff measures ${\mathcal{H}}_{\mathit{\mu }}^{q,t}$ and the relative packing measures ${\mathcal{P}}_{\mathit{\mu }}^{q,t}$ defined in a separable metric space. We give some product inequalities which are a consequence of a new version of density theorems for these measures. Moreover, we prove that ${\mathcal{H}}_{\mathit{\mu }}^{q,t}$ and ${\mathcal{P}}_{\mathit{\mu }}^{q,t}$ can be expressed as Henstock-Thomson variation measures. The question of the weak-Vitali property arises in this context.