度量测度空间通过Urysohn万有空间嵌入的收敛性
Convergence of metric measure spaces via embeddings in the Urysohn universal space
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中文总结 AI 辅助
本文通过将度量测度空间等距嵌入Urysohn万有空间，证明了Gromov盒拓扑与弱拓扑的商拓扑一致，并构造了完备可分的截断Wasserstein距离，与盒距离Hölder等价。
中文摘要 AI 辅助
我们通过等距嵌入到Urysohn万有度量空间$\mathbb U$中，研究了度量测度空间的不同收敛概念。由于$\mathbb U$的万有性，规范化度量测度空间的同构类集合$\mathbb X_1$可以典范地等同于$\mathbb U$上Borel概率测度空间$\mathscr P(\mathbb U)$的商集$\mathscr P_\sim(\mathbb U)=\mathscr P(\mathbb U)/\sim$，其中$\mu\sim\nu$当且仅当$\nu$是$\mu$在它们各自支撑集之间的等距映射下的前推。通过关键利用$\mathbb U$的超齐性，我们证明了在上述等同下，$\mathbb X_1$上的Gromov盒拓扑与由$\mathscr P(\mathbb U)$的弱拓扑诱导的商拓扑一致。更定量地，$\mathscr P(\mathbb U)$上的截断$1$-Wasserstein距离在$\mathbb X_1\cong\mathscr P_\sim(\mathbb U)$上诱导出一个完备且可分的距离${\sf d}_{\rm mG}$，该距离度量化了$\mathscr P_\sim(\mathbb U)$的商拓扑，并且与盒距离$\square$是Hölder等价的。
英文摘要
We study different notions of convergence of metric measure spaces by means of isometric embeddings into the Urysohn universal metric space $\mathbb U$. Due to the universality of $\mathbb U$, the collection $\mathbb X_1$ of isomorphism classes of normalised metric measure spaces can be canonically identified with the quotient (set) $\mathscr P_\sim(\mathbb U)=\mathscr P(\mathbb U)/\sim$ of the space $\mathscr P(\mathbb U)$ of Borel probability measures on $\mathbb U$, where $μ\simν$ if $ν$ is the pushforward of $μ$ under an isometry between their respective supports. By making crucial use of the ultrahomogeneity of $\mathbb U$, we show that, under the above identification, Gromov's box topology on $\mathbb X_1$ coincides with the quotient topology induced by the weak topology of $\mathscr P(\mathbb U)$. More quantitatively, the truncated $1$-Wasserstein distance on $\mathscr P(\mathbb U)$ induces a complete and separable distance ${\sf d}_{\rm mG}$ on $\mathbb X_1\cong\mathscr P_\sim(\mathbb U)$, which metrises the quotient topology of $\mathscr P_\sim(\mathbb U)$ and is Hölder equivalent to the box distance $\square$.
发表机构
- University of Jyvaskyla（于韦斯屈莱大学）
- University of Novi Sad（诺维萨德大学）
机构由 AI 辅助整理，请以论文原文为准。
