Besicovitch-Federer projection theorem for mappings having constant rank of the Jacobian matrix

Abstract

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 $\Sigma\subset\mathbb{R}^n$ having finite Hausdorff measure and $\varepsilon>0$, we prove that for a mapping $f\in\mathcal{C}^1(U,\mathbb{R}^n)$ having constant, equal to $m$, rank of the Jacobian matrix there exists a mapping $f_\varepsilon$ whose rank of the Jacobian matrix is also constant, equal to $m$, such that $\|f_\varepsilon-f\|_{\mathcal{C}^1}<\varepsilon$ and $\mathcal{H}^m(f_\varepsilon(\Sigma))=0$. We derive it as a consequence of the Besicovitch-Federer theorem stating that the $\mathcal{H}^m$ measure of a generic projection of an $m$-unrectifiable set $\Sigma$ onto an $m$-dimensional plane is equal to zero.