Geodesic
On the support of measures with fixed marginals with applications in optimal mass transportation
Canadian mathematical bulletin, Tome 67 (2024) no. 4, pp. 1059-1068

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DOI

Let $\mu $ and $\nu $ be Borel probability measures on complete separable metric spaces X and Y, respectively. Each Borel probability measure $\gamma $ on $X\times Y$ with marginals $\mu $ and $\nu $ can be described through its disintegration $\big ( \gamma _{x}\big )_{x \in X}$ with respect to the initial distribution $\mu .$ Assume that $\mu $ is continuous, i.e., $\mu \big (\{x\}\big )=0$ for all $x \in X.$ We shall analyze the structure of the support of the measure $\gamma $ provided $\text {card } \big (\mathrm{spt} (\gamma _{x}) \big )$ is finitely countable for $\mu $-a.e. $x\in X.$ We shall also provide an application to optimal mass transportation.
DOI : 10.4153/S0008439524000377
Mots-clés : Optimal transport, bi-stochastic measures, Choquet theory 
Moameni, Abbas. On the support of measures with fixed marginals with applications in optimal mass transportation. Canadian mathematical bulletin, Tome 67 (2024) no. 4, pp. 1059-1068. doi: 10.4153/S0008439524000377
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