A Note on Fractal Measures and Cartesian Product Sets

In this paper, we give a new product formula : Ht(E)Hs(F)≤c1Ht+s(E×F)≤c2Hs(E)Pt(F)≤c3Pt+s(E×F)≤c4Ps(E)Pt(F).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathsf {H}}^t(E) {\mathsf {H}}^s(F)\le c_1 {\mathsf {H}}^{t+s}(E\times F)\le & {} c_2 {\mathsf {H}}^s(E) {\mathsf {P}}^t(F)\\\le & {} c_3 {\mathsf {P}}^{t+s}(E\times F) \le c_4 {\mathsf {P}}^s(E) {\mathsf {P}}^t(F). \end{aligned}$$\end{document}where E⊆Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E\subseteq \mathbb {R}^d$$\end{document}, F⊆Rl\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\subseteq \mathbb {R}^l$$\end{document} , t,s≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t,s\ge 0$$\end{document} and Ht\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathsf {H}}^t$$\end{document} and Ps\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {P}}^s$$\end{document} denote, respectively, the lower and upper Hewitt–Stromberg measures. Using these inequalities, we give lower and upper bounds for the lower and upper Hewitt–Stromberg dimensions b(E×F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {b}}(E\times F)$$\end{document} and B(E×F)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf {B}}(E\times F)$$\end{document} in terms of the Hewitt–Strombeg dimensions of E and F.