Comparison of fractal measures  

作　　者：QU Yanhui,WEN Shengyou & WEN Zhiying Department of Mathematics,Tsinghua University,Beijing 100084,China Department of Mathematics,Hubei University, Wuhan 430062, China 

出　　处：《Science China Mathematics》2005年第11期1545-1553,共9页中国科学：数学（英文版）

摘　　要：In this paper, the relationship between the s-dimensional Hausdorff measures and the g-measures in Rd is discussed, where g is a gauge function which is equivalent to ts and 0 < s≤d. It shows that if s=d, then Hg = c1Hd, Cg = c2Cd and Pg = c3Pd on Rd, where constants c1, c2 and c3 are determined by where Wg, Cg and Pg are the g-Hausdorff, g-central Hausdorff and g-packing measures on Rd respectively. In the case 0<s<d, some examples are given to show that the above conclusion may fail. However, there is always some s-set F (?) Rd such that Hg|F=C1HS|F, Cg|F = c2Cs|F and Pg|F = c3Ps|F, where the constants c1, c2 and c3 depend not only on g and s, but also on F. A criterion is presented for judging whether an s-set has the above properties.In this paper, the relationship between the s-dimensional Hausdorff measures and the g-measures in Rd is discussed, where g is a gauge function which is equivalent to ts and O ＜ s ≤ d. It shows that if s= d, then Hg= c1Hd, Cg= c2Cd and Pg = c3Pd on Rd,where constants c1, c2 and c3 are determined byc1 = c2 = lim inf t→O g(t)/td and c3 = lim sup t→O g(t)/td,where Hg, Cg and Pg are the g-Hausdorff, g-central Hausdorff and g-packing measures on Rd respectively. In the case O＜s＜d, some examples are given to show that the above conclusion may fail. However, there is always some s-set F ( ) Rd such thatHg|F= c1Hs|F, Cg|F = c2Cs|F and Pg|F = c3Ps|F,where the constants c1, c2 and c3 depend not only on g and s, but also on F. A criterion is presented for judging whether an s-set has the above properties.

关 键 词：gauge function Hausdorff measure central Hausdorff measure packing measure. 

分 类 号：O174.12[理学—数学]