We study a class of countably-infinite-dimensional linear programs (CILPs) whose feasible sets are bounded subsets of appropriately defined spaces of measures. The optimal value, optimal points, and minimal points of these CILPs can be approximated by solving finite-dimensional linear programs. We show how to construct finite-dimensional programs that lead to approximations with easy-to-evaluate error bounds, and we prove that the errors converge to zero as the size of the finite-dimensional programs approaches that of the original problem. We discuss the use of our methods in the computation of the stationary distributions, occupation measures, and exit distributions of Markov~chains.
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