# ON CANTOR SETS AND PACKING MEASURES

• WEI, CHUN ;
• WEN, SHENG-YOU
• Published : 2015.09.30
• 63 7

#### 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

#### References

1. 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.
2. A. Dvoretzky, A note on Hausdorff dimension functions, Math. Pro. Camb. Phil. Soc. 44 (1948), 13-16. https://doi.org/10.1017/S0305004100023938
3. K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wily & Sons, 1990.
4. D. J. Feng, Comparing packing measures to Hausdorff measures on the line, Math. Nachr. 241 (2002), 65-72. https://doi.org/10.1002/1522-2616(200207)241:1<65::AID-MANA65>3.0.CO;2-I
5. 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. https://doi.org/10.1112/S0024609399006256
6. 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.
7. 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.
8. H. Joyce and D. Preiss, On the existence of subsets of finite positive packing measure, Mathematika 42 (1995), no. 1, 15-24. https://doi.org/10.1112/S002557930001130X
9. 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. https://doi.org/10.1017/S0305004100072224
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. https://doi.org/10.1017/S0305004100072789
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. https://doi.org/10.1007/s00605-010-0271-3
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. https://doi.org/10.1090/S0002-9947-1985-0776398-8
15. C. Tricot, Two definitions of fractional dimension, Math. Pro. Cambridge Philos. Soc. 91 (1982), no. 1, 57-74. https://doi.org/10.1017/S0305004100059119
16. S. Y. Wen and Z. Y. Wen, Some properties of packing measure with doubling gauge, Studia Math. 165 (2004), no. 2, 125-134. https://doi.org/10.4064/sm165-2-3
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. https://doi.org/10.1002/mana.200510671

#### Acknowledgement

Supported by : NSFC