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 Kaarik and Parna (Acta Appl. Math. 78, 175–183, 2003; Acta Comment. Univ. Tartu. 8, 101–112, 2004) where the case of parametric sets was studied.