On the study of multifractal analysis of functions in a metric space

Multi-fractal analysis plays a crucial role in understanding the complex behaviors of functions across different fields. In this study, we presented an innovative approach to examining the multi-fractal formalism. Specifically, we introduced new multi-fractal Hausdorff and packing measures, enabling the exploration of the multi-fractal spectrum within a metric space and offering a novel proof that extended the classical results in this setting. As an application, we focused on the Birkhoff averages when the multi-fractal formalism did not hold.