加倍度量与加倍测度
The doubling metric and doubling measures
摘要：
本文在度量空间的非空有界开子集集上引入了所谓的加倍度量。给定度量空间$X$的子集$U$，$U$的前身$U_{*}$通过将包含在$U$内的所有开球的半径加倍并取它们的并集来定义。如果$U$是开集，则$U$的前身是包含$U$的开集。$U$和另一子集$V$之间的定向加倍距离是需要对$U$应用前导操作以获得包含$V$的集合的次数。最后，$U$和$V$之间的加倍距离是$U$和$V$之间的有向距离和$V$和$U$之间的有向距离的最大值。
英文摘要：
We introduce the so--called doubling metric on the collection of non--empty bounded open subsets of a metric space. Given a subset $U$ of a metric space $X$, the predecessor $U_{*}$ of $U$ is defined by doubling the radii of all open balls contained inside $U$, and taking their union. If $U$ is open, the predecessor of $U$ is an open set containing $U$. The directed doubling distance between $U$ and another subset $V$ is the number of times that the predecessor operation needs to be applied to $U$ to obtain a set that contains $V$. Finally, the doubling distance between $U$ and $V$ is the maximum of the directed distance between $U$ and $V$ and the directed distance between $V$ and $U$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
