  We point out necessary and sufficient conditions of uniform consistency of
nonparametric sets of alternatives for widespread nonparametric tests.
Nonparametric sets of alternatives can be defined both in terms of distribution
function and in terms of density (or signals in the problem of signal detection
in Gaussian white noise). In this part of paper such conditions are provided
for $\chi^2-$tests with increasing number of cells, Cramer-von Mises tests,
tests generated $\mathbb{L}_2$- norms of kernel estimators and tests generated
quadratic forms of estimators of Fourier coefficients.
