On distinguishability of hypotheses

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 $L_2$.