Given a discrete-valued sample X 1,...,X n , we wish to decide whether it was generated by a distribution belonging to a family H 0, or it was generated by a distribution belonging to a family H 1. In this work we assume that all distributions are stationary ergodic and do not make any further assumptions (in particular, no independence or mixing rate assumptions). We find some necessary and some sufficient conditions, formulated in terms of the topological properties of H 0 and H 1, for the existence of a consistent test. For the case where H 1 is the complement of H 0 (to the set of all stationary ergodic processes), these necessary and sufficient conditions coincide, thereby providing a complete characterization of families of processes membership to which can be consistently tested, against their complement, based on sampling. This criterion includes as special cases several known and some new results on testing for membership to various parametric families, as well as testing identity, independence, and other hypotheses.