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On the support of measures with fixed marginals with applications in optimal mass transportation
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- Journal:
- Canadian Mathematical Bulletin / Volume 67 / Issue 4 / December 2024
- Published online by Cambridge University Press:
- 29 May 2024, pp. 1059-1068
- Print publication:
- December 2024
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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.
