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We study Luzin N-property with respect to the Hausdorff measures for Sobolevspaces W^k_p(R^n,R^d). We prove that such N-property holds except for onecritical dimensional value t_*=n-(k-1)p; for this critical value the N-propertyfails in general, and we constructed the corresponding nontrivialcounterexample (based on the theory of lacunary Fourier series). Nevertheless,this N-property holds if we assume in addition that the highest k-derivativesbelongs to the Lorentz space L_{p,1} instead of L_p.  We extend these results to the case of fractional Sobolev spaces as well.Also, we establish some Fubini type theorems for $N$-properties and discusstheir applications to the Morse--Sard theorem and its recent extensions.

On Luzin N-property and uncertainty principle for the Sobolev mappings