  We study Luzin N-property with respect to the Hausdorff measures for Sobolev
spaces W^k_p(R^n,R^d). We prove that such N-property holds except for one
critical dimensional value t_*=n-(k-1)p; for this critical value the N-property
fails in general, and we constructed the corresponding nontrivial
counterexample (based on the theory of lacunary Fourier series). Nevertheless,
this N-property holds if we assume in addition that the highest k-derivatives
belongs 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 discuss
their applications to the Morse--Sard theorem and its recent extensions.
