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...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.展开更多
文摘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.
