ON CANTOR SETS AND PACKING MEASURES

다국어 초록 (Multilingual Abstract)

For every doubling gauge g, we prove that there is a Cantor set of positive finite Hg-measure, Pg-measure, and Pg 0 -premeasure. Also, we show that every compact metric space of infinite Pg 0 -premeasure has a compact countable subset of infinite Pg 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 F = E∪F, and a doubling gauge g such that E ∪ F has different positive finite Pg-measure and Pg 0 -premeasure.

참고문헌 (Reference)

1 C. Tricot, "Two definitions of fractional dimension" 91 (91): 57-74, 1982

2 Y. Peres, "The self-affine carpets of McMullen and Bedford have infinite Hausdorff mea-sure" 116 (116): 513-526, 1994

3 Y. Peres, "The packing measure of self-affine carpets" 115 (115): 437-450, 1994

4 D. J. Feng, "Some relations between packing pre-measure and packing measure" 31 (31): 665-670, 1999

5 S. Y. Wen, "Some properties of packing measure with doubling gauge" 165 (165): 125-134, 2004

6 D. J. Feng, "Some dimensional results for homogeneous Moran sets" 40 (40): 475-482, 1997

7 S. J. Taylor, "Packing measure and its evaluation for a Brownian path" 288 (288): 679-699, 1985

8 H. Joyce, "On the existence of subsets of finite positive packing measure" 42 (42): 15-24, 1995

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

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

11 S. Y. Wen, "Gauges for the self-similar sets" 281 (281): 1205-1214, 2008

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

13 D. J. Feng, "Dimensions and gauges for symmetric cantor sets" 11 (11): 108-110, 1995

14 T. Rajala, "Comparing the Hausdorff and packing measures of sets of small dimension in metric spaces" 164 (164): 313-323, 2011

15 D. J. Feng, "Comparing packing measures to Hausdorff measures on the line" 241 : 65-72, 2002

16 M. Cs¨ornyei, "An example illustrating Pg(K) ≠ Pg0(K) for sets of finite pre-measure" 27 (27): 65-70, 2001

17 A. Dvoretzky, "A note on Hausdorff dimension functions" 44 : 13-16, 1948