## Publicacions Matemàtiques

### Packing Measures on Cartesian Products

Ondřej Zindulka

#### 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

https://projecteuclid.org/euclid.pm/1386857701

Mathematical Reviews number (MathSciNet)
MR3114775

Zentralblatt MATH identifier
1285.28009

#### Citation

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

#### References

• P. Assouad, Étude d'une dimension métrique liée à la possibilité de plongements dans $\mathbf{R}^{n}$, C. R. Acad. Sci. Paris Sér. A-B 288(15) (1979), A731\NdashA734.
• C. J. Bishop and Y. Peres, Packing dimension and Cartesian products, Trans. Amer. Math. Soc. 348(11) (1996), 4433\Ndash4445. \small\tt DOI: 10.1090/S0002-9947-96-01750-3.
• D.-C. Chang and Y. Xu, New inequalities on fractal analysis and their applications, J. Inequal. Appl. Art. ID 26249 (2007), 17 pp. \small\tt DOI: 10.1155/2007/26249.
• M. Das, Equality of the packing and pseudo-packing measures, Bull. Polish Acad. Sci. Math. 49(1) (2001), 73\Ndash79.
• G. A. Edgar, Packing measure in general metric space, Real Anal. Exchange 26(2) (2000/01), 831\Ndash852.
• G. A. Edgar, Centered densities and fractal measures, New York J. Math. 13 (2007), 33\Ndash87 (electronic).
• G. A. Edgar, “Integral, probability, and fractal measures”, Springer-Verlag, New York, 1998.
• H. Haase, A contribution to measure and dimension of metricspaces, Math. Nachr. 124 (1985), 45\Ndash55. \small\tt DOI: 10.1002/mana. \small\tt 19851240104.
• H. Haase, Open-invariant measures and the covering number ofsets, Math. Nachr. 134 (1987), 295\Ndash307. \small\tt DOI: 10.1002/mana. \small\tt 19871340121.
• E. Hewitt and K. Stromberg, “Real and abstract analysis. A modern treatment of the theory of functions of a real variable”, Springer-Verlag, New York, 1965.
• J. D. Howroyd, On Hausdorff and packing dimension of product spaces, Math. Proc. Cambridge Philos. Soc. 119(4) (1996), 715\Ndash727. \small\tt DOI: 10.1017/S0305004100074545.
• H. Joyce and D. Preiss, On the existence of subsets of finite positive packing measure, Mathematika 42(1) (1995), 15\Ndash24. \small\tt DOI: 10.1112/S002557930001130X.
• D. G. Larman, A new theory of dimension, Proc. London Math. Soc. (3) 17 (1967), 178\Ndash192. \small\tt DOI: 10.1112/plms/s3-17.1.178.
• J. Luukkainen, Assouad dimension: antifractal metrization, porous sets, and homogeneous measures J. Korean Math. Soc. 35(1) (1998), 23\Ndash76.
• P. Mattila, “Geometry of sets and measures in Euclidean spaces”, Fractals and rectifiability, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, Cambridge, 1995.
• Y. B. Pesin, “Dimension theory in dynamical systems”, Contemporary views and applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.
• C. A. Rogers, “Hausdorff measures”, Cambridge University Press, London-New York, 1970.
• C. Tricot, Jr., Two definitions of fractional dimension, Math.Proc. Cambridge Philos. Soc. 91(1) (1982), 57\Ndash74. \small\tt DOI: 10.1017/ \small\tt S0305004100059119.
• Y. Xiao, Packing dimension, Hausdorff dimension and Cartesian product sets, Math. Proc. Cambridge Philos. Soc. 120(3) (1996), 535\Ndash546. \small\tt DOI: 10.1017/S030500410007506X.