Hausdorff dimension and doubling measures on metric spaces
Hausdorff dimension and doubling measures on metric spaces
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DOI：
10.1090/s0002-9939-98-04317-2
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发表时间：
1998
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通讯作者：
J. Wu
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作者：
J. Wu
贝格和Konyagin证明了紧度量空间具有非平凡的加倍测度当且仅当它具有有限的一致度量维数。他们的加倍措施的建设需要无限多的调整。给出了一个更简单、更直接的构造，并证明了对任意α > 0，在Hausdorff维数不超过α的集合上，可以选择具有满测度的加倍测度.设（X，ρ）是紧度量空间.贝格和Konyagin在[VK]中证明了（X，ρ）有非平凡的加倍测度μ（存在Λ ≥ 1使得μ（B（x，2 r））≤ Λμ（B（x，r））当且仅当（X，ρ）有有限的一致度量维数（在每个球B（x，2 r）中，至多存在N个点，它们的相互距离至少为r）.这里B（x，r）= {y：ρ（x，y）0，存在X上的一个加倍测度，它在Hausdorff维数的集合上的满测度至多为α.我们还观察到，即使X是正长的真实的直线上的集合，加倍测度也可以集中在可数集上。有些观点是从[FKP]、[VK]和[T]改编而来的。1.定理与例子设（X，ρ）是一个具有有限一致度量维数的紧致度量空间，且直径X < 1.对于每个k ≥ 0，令Sk = {xk，j：1 ≤ j ≤ J（k）}是X上的极大10− k-网（Sk中的点相互距离至少为10 −k，Sk外的点到Sk的距离小于10−k），满足S 0 <$S1 <$··<$Sk <$Sk+1 <$··<$。注意，S 0只有一个点x 0，1。对于每个k ≥ 0，令{Tk，j：1 ≤ j ≤ J（k）}是Sk+1的一个划分，满足Sk+1 <$B（xk，j，10−k/2）<$Tk，j <$Sk+1 <$B（xk，j，10−k）。(1.1)编辑于1996年10月24日收到。1991年数学学科分类小学28 C15;中学54 E35，54 E45。
Vol′berg and Konyagin have proved that a compact metric space carries a nontrivial doubling measure if and only if it has finite uniform metric dimension. Their construction of doubling measures requires infinitely many adjustments. We give a simpler and more direct construction, and also prove that for any α > 0, the doubling measure may be chosen to have full measure on a set of Hausdorff dimension at most α. Let (X, ρ) be a compact metric space. Vol’berg and Konyagin proved in [VK] that (X, ρ) carries a nontrivial doubling measure μ (there exists Λ ≥ 1 so that μ(B(x, 2r)) ≤ Λμ(B(x, r)) for all x ∈ X and r > 0) if and only if (X, ρ) has finite uniform metric dimension (in each ball B(x, 2r), there exist at most N points with mutual distances at least r). Here B(x, r) = {y : ρ(x, y) 0, there exists a doubling measure on X that has full measure on a set of Hausdorff dimension at most α. Also we observe that a doubling measure may be concentrated on a countable set even when X is a set on the real line of positive length. Some ideas have been adapted from [FKP], [VK] and [T]. 1. Theorems and examples Assume, from now on, that (X, ρ) is a compact metric space of finite uniform metric dimension and that diam X < 1. For each k ≥ 0, let Sk = {xk,j : 1 ≤ j ≤ J(k)} be a maximal 10−k-net on X (points in Sk having mutual distances at least 10 −k, and points outside Sk having distances less than 10−k to Sk), satisfying S0 ⊆ S1 ⊆ · · · ⊆ Sk ⊆ Sk+1 ⊆ · · · . Note that S0 has only one point x0,1. For each k ≥ 0, let {Tk,j : 1 ≤ j ≤ J(k)} be a partition of Sk+1 satisfying Sk+1 ∩B(xk,j , 10−k/2) ⊆ Tk,j ⊆ Sk+1 ∩B(xk,j , 10−k). (1.1) Received by the editors October 24, 1996. 1991 Mathematics Subject Classification. Primary 28C15; Secondary 54E35, 54E45.