ON CANTOR SETS AND PACKING MEASURES

WEI, CHUN; WEN, SHENG-YOU

doi:10.4134/BKMS.2015.52.5.1737

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ON CANTOR SETS AND PACKING MEASURES

Journal title : Bulletin of the Korean Mathematical Society
Volume 52, Issue 5, 2015, pp.1737-1751
Publisher : The Korean Mathematical Society
DOI : 10.4134/BKMS.2015.52.5.1737

Title & Authors

ON CANTOR SETS AND PACKING MEASURES
WEI, CHUN; WEN, SHENG-YOU;

Abstract

For every doubling gauge g, we prove that there is a Cantor set of positive finite ${$H^g$}$ -measure, ${$P^g$}$ -measure, and ${$P^g_0$}$ -premeasure. Also, we show that every compact metric space of infinite ${$P^g_0$}$ -premeasure has a compact countable subset of infinite ${$P^g_0$}$ -premeasure. In addition, we obtain a class of uniform Cantor sets and prove that, for every set E in this class, there exists a countable set F, with ${$\bar{F}=E{\cup}F$}$ , and a doubling gauge g such that ${$E{\cup}F$}$ has different positive finite ${$P^g$}$ -measure and ${$P^g_0$}$ -premeasure.

Keywords

Cantor set;packing measure;premeasure;gauge function;doubling condition;

Language

English

Cited by

References

M. Csornyei, An example illustrating $P^g$(K) $\neq$ $P^g_0$ (K) for sets of finite pre-measure, Real Anal. Exchange 27 (2001/02), no. 1, 65-70.

A. Dvoretzky, A note on Hausdorff dimension functions, Math. Pro. Camb. Phil. Soc. 44 (1948), 13-16.

K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wily & Sons, 1990.

D. J. Feng, Comparing packing measures to Hausdorff measures on the line, Math. Nachr. 241 (2002), 65-72.

D. J. Feng, S. Hua, and Z. Y. Wen, Some relations between packing pre-measure and packing measure, Bull. London Math. Soc. 31 (1999), no. 6, 665-670.

D. J. Feng, H. Rao, Z. Y. Wen, and J. Wu, Dimensions and gauges for symmetric cantor sets, Approx. Theory Appl. 11 (1995), no. 4, 108-110.

D. J. Feng, W. Wen, and J. Wu, Some dimensional results for homogeneous Moran sets, Sci. China Ser. A 40 (1997), no. 5, 475-482.

H. Joyce and D. Preiss, On the existence of subsets of finite positive packing measure, Mathematika 42 (1995), no. 1, 15-24.

P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995.

10.

Y. Peres, The packing measure of self-affine carpets, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 3, 437-450.

11.

Y. Peres, The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure, Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 3, 513-526.

12.

T. Rajala, Comparing the Hausdorff and packing measures of sets of small dimension in metric spaces, Monatsh Math. 164 (2011), no. 3, 313-323.

13.

C. Rogers, Hausdorff Measures, Cambridge University Press, 1998.

14.

S. J. Taylor and C. Tricot, Packing measure and its evaluation for a Brownian path, Trans. Amer. Math. Soc. 288 (1985), no. 2, 679-699.

15.

C. Tricot, Two definitions of fractional dimension, Math. Pro. Cambridge Philos. Soc. 91 (1982), no. 1, 57-74.

16.

S. Y. Wen and Z. Y. Wen, Some properties of packing measure with doubling gauge, Studia Math. 165 (2004), no. 2, 125-134.

17.

S. Y. Wen, Z. X. Wen, and Z. Y. Wen, Gauges for the self-similar sets, Math. Nachr. 281 (2008), no. 8, 1205-1214.