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Packing Measures on Cartesian Products

Ondřej Zindulka

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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

Article information

Publ. Mat., Volume 57, Number 2 (2013), 393-420.

First available in Project Euclid: 12 December 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A78: Hausdorff and packing measures 28A80: Fractals [See also 37Fxx] 54E35: Metric spaces, metrizability

Packing measure lower packing measure packing dimension lower packing dimension cartesian product


Zindulka, Ondřej. Packing Measures on Cartesian Products. Publ. Mat. 57 (2013), no. 2, 393--420.

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