Continuous dependence of sets in a space of measures and a program minimax problem

For conflict-controlled dynamical systems satisfying the conditions of generalized uniqueness and uniform boundedness, the solvability of the minimax problem in the class of generalized controls is studied. The issues of consistency of such an extension are considered; i. e., the possibility of approximating generalized controls in the space of strategic measures by embeddings of ordinary controls is analyzed. For this purpose, the dependence of the set of measures on the general marginal distribution specified on one of the factors of the base space is studied. The continuity of this dependence in the Hausdorff metric defined by the metric corresponding to the $*$-weak topology in the space of measures is established. The density of embeddings of ordinary controls and control-noise pairs in sets of corresponding generalized controls in the $*$-weak topologies is also shown.