AI 中文总结
该研究解决了Aldous等人的猜想，证明无原子Borel概率测度可作为完备可分度量空间上两点距离分布的充要条件，且实现度量与超度量L-双Lipschitz等价。
AI 中文摘要
我们研究[0,∞)上的哪些概率测度可作为从配备Borel概率测度的完备可分度量空间中抽取的两个独立点之间的距离分布。我们证明，无原子Borel概率测度可通过这种方式实现当且仅当其支撑集包含原点。这解决了Aldous、Blanc和Curien关于绝对连续律的猜想，也涵盖了相对于Lebesgue测度奇异的无原子测度。此外，对任意L>1，我们证明实现的度量可选取为与超度量L-双Lipschitz等价。该证明从紧致分量构造空间，分量间的距离生成测度的规定部分，而分量内的距离则分配给更小尺度。
英文摘要
We study which probability measures on $[0,\infty)$ can occur as the distribution of the distance between two independent points sampled from a complete separable metric space equipped with a Borel probability measure. We prove that an atomless Borel probability measure can be realised in this way if and only if its support contains the origin. This settles the conjecture of Aldous, Blanc, and Curien for absolutely continuous laws and also covers atomless measures that are singular with respect to Lebesgue measure. Moreover, for every $L>1$, we show that the realising metric may be chosen $L$-bi-Lipschitz equivalent to an ultrametric. The proof constructs the space from compact components whose mutual distances produce prescribed parts of the measure, while distances within the components are assigned to smaller scales.
