  The paper addresses a problem of sampling discretization of integral norms of
elements of finite-dimensional subspaces satisfying some conditions. We prove
sampling discretization results under a standard assumption formulated in terms
of the Nikol'skii-type inequality. {In particular, we obtain} some upper bounds
on the number of sample points sufficient for good discretization of the
integral $L_p$ norms, $1\le p<2$, of functions from finite-dimensional
subspaces of continuous functions. Our new results improve upon the known
results in this direction. We use a new technique based on deep results of
Talagrand from functional analysis.
