  We consider the problems of hypothesis testing on a probability measure of
independent sample, on solution of ill-posed problem, on deconvolution problem
and on Poisson mean measure. For all these setups necessary conditions and
sufficient conditions are given for distinguishability of sets of hypothesis.
In the case of hypothesis testing on a probability measure and on Poisson mean
measure the results are given in terms of weak topology and topology of weak
convergence on all Borel sets.
  The problem of discernibility of hypothesis is also studied. In other cases
the necessary and sufficient conditions of distinguishability are given if the
sets of hypotheses are bounded sets in $L_2$.
