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Regular version of the site

Article

Kantorovich problems and conditional measures depending on a parameter

Journal of Mathematical Analysis and Applications. 2020. Vol. 486. No. 1 (123883). P. 1-30.

We study measurable dependence of measures on a parameter in the following two classical problems: constructing conditional measures and the Kantorovich optimal transportation. For parametric families of measures and mappings we prove the existence of conditional measures measurably depending on the parameter. A particular emphasis is made on the Borel measurability (which cannot be always achieved). Our second main result gives sufficient conditions for the Borel measurability of optimal transports and transportation costs with respect to a parameter in the case where marginal measures and cost functions depend on a parameter. As a corollary we obtain the Borel measurability with respect to the parameter for disintegrations of optimal plans. Finally, we show that the Skorohod parametrization of measures by mappings can be also made measurable with respect to a parameter.