Nonlinear Kantorovich problems with a parameter

We consider nonlinear Kantorovich problems with marginal distributions and cost functions depending measurably on a parameter and prove that there exist optimal transportation plans that are also measurable with respect to the parameter. Unlike the classical linear Kantorovich problem of minimization of the integrals of a given cost function with respect to transportation plans, we deal with nonlinear cost functionals in which integrands depend on transportation plans. Dependence of cost functions on conditional measures of transportation plans is also allowed.