Publicacions Matemàtiques

Packing Measures on Cartesian Products

Ondřej Zindulka

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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

Article information

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

Dates
First available in Project Euclid: 12 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.pm/1386857701

Mathematical Reviews number (MathSciNet)
MR3114775

Zentralblatt MATH identifier
1285.28009

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

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

Citation

Zindulka, Ondřej. Packing Measures on Cartesian Products. Publ. Mat. 57 (2013), no. 2, 393--420. https://projecteuclid.org/euclid.pm/1386857701


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