  This paper explores conditions of existence of different types of consistent
tests. New links of these types of consistency are also established. The
existence of discernible (strong consistent) tests follows from the existence
of pointwise consistent tests. If there are consistent tests, then the set of
alternatives can be represented as countable union of nested subsets such that
there are uniformly consistent tests for the hypothesis and each of this subset
of alternatives. Implementing these results we explore both sufficient
conditions and necessary conditions for existence of consistent, pointwise
consistent, uniformly consistent and discernible tests for hypothesis testing
on a probability measure of independent sample, on a mean measure of Poisson
process, on a solution of linear ill-posed problems in Gaussian noise, on a
solution of deconvolution problem and for the problem of signal detection in
Gaussian white noise. In the last three cases the necessary conditions and
sufficient conditions coincide.
