A criterion for pure unrectifiability of sets (via universal vector bundle). (English) Zbl 1234.28007
Summary: Let \(m,n\) be positive integers such that \(m<n\) and let \(G(n,m)\) be the Grassmann manifold of all \(m\)-dimensional subspaces of \(\mathbb{R}^n\). For \(V\in G(n,m)\) let \(\pi_V\) denote the orthogonal projection from \(\mathbb{R}^n\) onto \(V\). The following characterization of purely unrectifiable sets holds. Let \(A\) be an \({\mathcal H}^m\)-measurable subset of \(\mathbb{R}^n\) with \({\mathcal H}^m(A)<\infty\). Then \(A\) is purely \(m\)-unrectifiable if and only if there exists a null subset \(Z\) of the universal bundle \(\{(V,v)|V\in G(n,m),v\in V\}\) such that, for all \(P\in A\), one has \({\mathcal H}^{m(n-m)}(\{V\in G(n,m)|(V,\pi_V(P))\in Z\})>0\). One can replace “for all \(P\in A\)” by “for \({\mathcal H}^m\)-a.e. \(P\in A\)”.
MSC:
28A75 | Length, area, volume, other geometric measure theory |
28A78 | Hausdorff and packing measures |
49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |
53A05 | Surfaces in Euclidean and related spaces |