LAUSR

The purpose of this article is to prove a generalisation of the Besicovitch–Federer projection theorem about a characterisation of rectifiable and unrectifiable sets in terms of their projections. For an m-unrectifiable set $$\varSigma \subset \mathbb {R}^{n}$$Σ⊂Rn having finite Hausdorff measure and $$\varepsilon >0$$ε>0, we prove that for a mapping $$f\in \mathscr {C}^1(U,\mathbb {R}^{n})$$f∈C1(U,Rn) having constant, equal to m, rank of the Jacobian matrix there exists a mapping $$f_\varepsilon $$fε whose rank of the Jacobian matrix is also constant, equal to m, such that $$\Vert f_\varepsilon -f\Vert _{\mathscr {C}^1}<\varepsilon $$‖fε-f‖C1<ε and $${\mathscr {H}}^{m}(f_\varepsilon (\varSigma ))=0$$Hm(fε(Σ))=0. We derive it as a consequence of the Besicovitch–Federer theorem stating that the $${\mathscr {H}}^m$$Hm measure of a generic projection of an m-unrectifiable set $$\varSigma $$Σ onto an m-dimensional plane is equal to zero.