We study measurable dependence of measures on a parameter in the following two classical problems: constructing conditional measures and the Kantorovich optimal transportation. We obtain broad sufficient conditions for the existence of conditional probabilities measurably depending on a parameter in the case of parametric families of measures and mappings. 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 conditional measures of optimal plans. Finally, we show that the Skorohod parametrization of measures by mappings can be also made measurable with respect to a parameter.

中文翻译：

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Kantorovich 问题和取决于参数的条件测度

我们研究了以下两个经典问题中措施对参数的可测量依赖性：构造条件措施和 Kantorovich 最优运输。在度量和映射的参数族的情况下，我们根据一个参数获得了条件概率的存在的广泛充分条件。特别强调了 Borel 可测量性（这不能总是实现）。在边际度量和成本函数取决于参数的情况下，我们的第二个主要结果为最佳运输和运输成本相对于参数的 Borel 可测量性提供了充分条件。作为推论，我们获得了关于最优计划的条件测度参数的 Borel 可测性。最后，