We say that a mapping v: ℝn → ℝd satisfies the (τ, σ ) -N-property if Hσ( v (E)) = 0 whenever Hτ (E) = 0, where Hτ means the Hausdorff measure. We prove that every mapping v of Sobolev class W p k (ℝn,ℝd ) with kp > n satisfies the (τ, σ )-N-property for every 0 < τ ≠ τ*: = n - (k -1)p with We prove also that for k > 1 and for the critical value τ = τ* the corresponding (τ, σ )-N-property fails in general. Nevertheless, this (τ, σ )-N-property holds for τ = τ* if we assume in addition that the highest derivatives ∇kv belong to the Lorentz space Lp,1(ℝn ) instead of Lp. We extend these results to the case of fractional Sobolev spaces as well. Also, we establish some Fubini-type theorems for N-Nproperties and discuss their applications to the Morse-Sard theorem and its recent extensions.