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We study measurable dependence of measures on a parameter in the following two classical problems: constructing conditional measures and the Kantorovich optimal transportation.
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A.N. Doledenok, On a Kantorovich problem with a density constraint. Mat. Zametki 104 (1) (2018), 45–55 (in Russian); English transl.: Math. Notes 104 (1) (2018), 39–47
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