Attainability property and ergodicity theorems for a semi-Markov process with an arbitrary set of states

The connection is studied which exists between the finiteness of the average time to achieve certain subsets, on the one hand, and, on the other, the ergodic properties of the process. The assertions proven in this paper generalize results obtained for semi-Markov processes with discrete sets of states. Moreover, the connection is studied between the asymptotic time to achieve a set whose “dimensions” tend to zero and, on the other hand, the ergodicity of the process.