  The paper studies the sampling discretization problem for integral norms on
subspaces of $L^p(\mu)$. Several close to optimal results are obtained on
subspaces for which certain Nikolskii-type inequality is valid. The problem of
norms discretization is connected with the probabilistic question about the
approximation with high probability of marginals of a high dimensional random
vector by sampling. As a byproduct of our approach we refine the result of O.
Gu$\acute{e}$don and M. Rudelson concerning the approximation of marginals. In
particular, the obtained improvement recovers a theorem of J. Bourgain, J.
Lindenstrauss, and V. Milman concerning embeddings of finite dimensional
subspaces of $L^p[0, 1]$ into $\ell_p^m$. The proofs in the paper use the
recent developments of the chaining technique by R. van Handel.
