Approximation of Distributions by Bounded Sets

Let P be a probability distribution on a locally compact separable metric space (S,d). We study the following problem of approximation of a distribution P by a set A from a given class $\mathcal{A}\subset2^{S}$ : $$W(A,P)\equiv\int_{S}\varphi(d(x,A))P(dx)\to\min_{A\in\mathcal{A}},$$ where φ is a nondecreasing function. A special case where $\mathcal{A}$ consists of unions of bounded sets, $\mathcal{A}=\{\bigcup_{i=1}^{k}A_{i}:\Delta(A_{i})\leq K,\ i=1,\ldots,k\}$ , is considered in detail. We give sufficient conditions for the existence of an optimal approximative set and for the convergence of the sequence of optimal sets A _n found for measures P _n which satisfy P _n ? P. Current article is a follow-up to Käärik and Pärna (Acta Appl. Math. 78, 175–183, 2003; Acta Comment. Univ. Tartu. 8, 101–112, 2004) where the case of parametric sets was studied.