Packing Measures on Cartesian Products

Abstract

Packing measures $\mathscr{P}^{g}(E)$ and Hewitt-Stromberg measures $\boldsymbol{\nu}^{g}(E)$ and their relatives are investigated. It is shown, for instance, that for any metric spaces $X$,~$Y$ and any Hausdorff functions $f$, $g$ $$ \boldsymbol{\nu}^{g}(X)\cdot\mathscr{P}^{h}(Y)\leqslant\mathscr{P}^{gh}(X\times Y). $$ The inequality for the corresponding dimensions is established and used for a solution of a problem of Hu and Taylor: If $X\subseteq\mathbb{R}^n$, then $$ \inf\{\overline{\dim}_{\mathsf{P}} X\times Y-\overline{\dim}_{\mathsf{P}} Y:Y\subseteq\mathbb{R}^n\} =\liminf_{X_n{\nearrow} X}\underline{\dim}_{\mathsf{B}} X_n. $$ Corresponding dimension inequalities for products of measures are established