Sets that Support a Joint Distribution

Abstract

Given probability distributions $\mu$ and $\nu$ 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 $\mu$ and $\nu$, 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 $\mu$ and $\nu$ 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).

22 pages, 5 figures