Existence of consistent test sequences at the complex null hypotheses in discrete statistical problems

The problem of existence of a consistent test sequence is considered for testing of complex hypothesis against complex alternatives in a sequence of finite spaces. When the sequence of spaces is generated by Cartesian product of a finite set and probability measures on these spaces are consistent, it is possible to find sufficient conditions of existence of consistent test sequence in terms of topological properties of the certain sets. Under additional conditions it is possible to refuse the requirement of domination of certain measures from a null hypothesis and uniform limitation of density.