Approximation of Distributions by Parametric Sets

Let P be a probability distribution on a separable metric space (S,d). We study the following problem of approximation of a distribution P by a set from a given class A2^S: W(A,P)_S(d(x,A))P(dx)min_AA, where is a nondecreasing function. A special case where A is a parametric class A={A():T} is considered in detail. Our main interest is to obtain convergence results for sequences {A^*_n}, where A^*_n is an optimal set for a measure P_n satisfying P_nP, as n.