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

Some improved bounds in sampling discretization of integral norms