We study numerical integration based on Markov chains. Our focus is on establishing error bounds uniformly on classes of integrands. Since in general state space the concept of uniform ergodicity is too restrictive to cover important cases, we analyze the error of V-uniformly ergodic Markov chains. We place emphasis on the interplay between ergodicity properties of the transition kernel, the initial distributions and the classes of integrands. Our analysis is based on arguments from interpolation theory.
