Probabilistic spaces and generalized dimensions: A multifractal approach

Consider a probability space (Z,ℱ,τ). This paper primarily investigates a general multifractal formalism within the probability space (Z,ℱ,τ). Our first objective is to introduce a multifractal generalization of the Hausdorff and packing measures. We then explore the properties of the general multifractal Hausdorff measure and the multifractal packing measure within (Z,ℱ,τ), examining their implications for the general multifractal spectrum functions. We investigate the relationship between the general multifractal measures and the nature of general multifractal dimensions within this framework. Additionally, we obtain an analogue of Frostman’s lemma for the general multifractal Hausdorff and packing measures in probability spaces. Using this analogue, we derive representations for the functions bℋπ̃ and bPπ̃. Furthermore, we provide a technique to demonstrate that E is an (α,π)-fractal with respect to τ, leading to density theorems for the multifractal Hausdorff and packing measures in these probability spaces. Finally, we present a general theorem for multifractal formalism on probability spaces, deriving results for general multifractal Hausdorff and packing functions that vary with respect to arbitrary probability measures at points α where the multifractal functions bℋπ̃(α) and bPπ̃(α) differ.