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The paper studies the sampling discretization problem for integral norms onsubspaces of $L^p(\mu)$. Several close to optimal results are obtained onsubspaces for which certain Nikolskii-type inequality is valid. The problem ofnorms discretization is connected with the probabilistic question about theapproximation with high probability of marginals of a high dimensional randomvector 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. Inparticular, the obtained improvement recovers a theorem of J. Bourgain, J.Lindenstrauss, and V. Milman concerning embeddings of finite dimensionalsubspaces of $L^p[0, 1]$ into $\ell_p^m$. The proofs in the paper use therecent developments of the chaining technique by R. van Handel.

Marcinkiewicz-type discretization of $L^p$-norms under the Nikolskii-type inequality assumption