支撑联合分布的集合
Sets that Support a Joint Distribution
AI总结：
该论文研究测度空间上联合分布支撑集的存在性问题，针对离散与连续边缘分布情形给出解答，将离散情形推广至图的流问题，对连续正则集给出简洁刻画。
AI中文摘要：
给定测度空间X和Y上的概率分布μ与ν，以及闭集S⊆X×Y，是否存在X×Y上的概率分布，其边缘分布为μ和ν，且支撑集恰好为S？我们针对边缘分布为离散分布、以及边缘分布为实数轴上连续分布的情形解答该问题。特别受关注的情形是S⊆[0,1]^2且μ与ν为勒贝格测度；此时上述问题等价于“S何时是双随机测度的支撑集”。离散情形被推广以判定（可能为无限的）边容量、节点加权图是否支撑满的无处零流；对于连续情形，我们在该集合为正则集（即其内部的闭包）时给出了特别简洁的刻画。
英文摘要：
Given probability distributions $μ$ and $ν$ on measure spaces $X$ and $Y$, and a closed set $S \subseteq X \times Y$, when is there a probability distribution on $X \times Y$ whose marginals are $μ$ and $ν$, and whose support is precisely $S$? We answer the question when the marginals are discrete, and when the marginals are continuous distributions on the real line. Of special interest is the case where $S \subseteq [0,1]^2$ and $μ$ and $ν$ are Lebesgue measure; then the above question is tantamount to ``when is $S$ the support of a doubly stochastic measure?". The discrete case is generalized to determine when a (possibly infinite) edge-capacitated, node-weighted graph supports a full, nowhere-zero flow; for the continuous case we provide a particularly straightforward characterization when the set in question is regular (i.e., is the closure of its interior).
