Some properties of new general fractal measures

In this research, we adopt a comprehensive approach to address the multifractal and fractal analysis problem. We introduce a novel definition for the general Hausdorff and packing measures by considering sums involving certain functions and variables. Specifically, we explore the sums of the form

$$\begin{aligned} \sum \limits _i h^{-1}\Big (q h\big (\mu \bigl (B(x_i,r_i)\bigl )\big )+tg(r_i)\Big ), \end{aligned}$$

where \(\mu \) represents a Borel probability measure on \(\mathbb R^d\), and q and t are real numbers. The functions h and g are predetermined and play a significant role in our proposed intrinsic definition. Our observation reveals that estimating Hausdorff and packing pre-measures is significantly easier than estimating the exact Hausdorff and packing measures. Therefore, it is natural and essential to explore the relationships between the Hausdorff and packing pre-measures and their corresponding measures. This investigation constitutes the primary objective of this paper. Additionally, the secondary aim is to establish that, in the case of finite pre-measures, they possess a form of outer regularity in a metric space X that is not limited to a specific context or framework.