@article{Attia2023, 
author = {Najmeddine Attia and Bilel Selmi},
title = {Different types of multifractal measures in separable metric spaces and their applications},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {6},
pages = {12889-12921},
keywords = {generalized Hausdorff measure, generalized packing measure, Henstock-Thomson variation, regularities, densities, doubling measures},
url = {https://www.sciopen.com/article/10.3934/math.2023650},
doi = {10.3934/math.2023650},
abstract = {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                      H                            μ                    q      ,      t       and the relative packing measures                      P                            μ                    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                      H                            μ                    q      ,      t       and                      P                            μ                    q      ,      t       can be expressed as Henstock-Thomson variation measures. The question of the weak-Vitali property arises in this context.}
}