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On the support of measures with fixed marginals with applications in optimal mass transportation

Part of: Manifolds

Published online by Cambridge University Press:  29 May 2024

Abbas Moameni
Abbas Moameni*
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada
*
e-mail: momeni@math.carleton.ca
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Abstract

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.

Keywords

Optimal transportbi-stochastic measuresChoquet theory

MSC classification

Primary: 49Q20: Variational problems in a geometric measure-theoretic setting
Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 4 , December 2024 , pp. 1059 - 1068
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This work is supported by a grant from the Natural Sciences and Engineering Research Council of Canada.

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