Advances on fractal measures of Cartesian product sets in Euclidean space

Let $\mu$ and $\nu$ be two compactly supported Borel probability measures on ${\mathbb{R}}^d$ and ${\mathbb{R}}^l$, respectively, and let $q\in \mathbb{R}$ and $h,g$ be two Hausdorff functions. In this paper, we are concerned with evaluation of the lower and upper Hewitt-Stromberg measure of Cartesian product sets, denoted, respectively, by ${\mathsf{H}}_{\mu }^{q,g}$ and ${\mathsf{P}}_{\nu }^{q,h}$, by means of the measure of their components. This is done by the construction of new multifractal measures in a similar manner to Hewitt-Stomberg measures but using the class of all (semi-) half-open binary cubes of covering sets in the definition rather than the class of all balls. Our derived product formula excludes the $0$– $\mathrm{\infty }$ case, and our approach is uniquely applied within an Euclidean space, distinguishing it from those previously utilized in metric spaces. Furthermore, by examining the measures of symmetric generalized Cantor sets, we establish that the exclusion of the $0$– $\mathrm{\infty }$ condition is essential and cannot be omitted.