  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.
