A criterion for pure unrectifiability of sets (via universal vector bundle)

Silvano Delladio¹

¹Dipartimento di Matematica Università di Trento Trento, Italy

Annales Polonici Mathematici, Tome 102 (2011) no. 1, pp. 73-78

Résumé

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 $\{ (V,v) \mid V\in G(n,m),\, v\in V\}$ such that, for all $P\in A$, one has $\mathcal H^{m(n-m)}(\{ V\in G(n,m) \mid (V,\pi_V(P))\in Z\})>0$. One can replace “for all $P\in A$” by “for $\mathcal H^m$-a.e. $P\in A$”.

Détail

DOI : 10.4064/ap102-1-6

Keywords: positive integers grassmann manifold m dimensional subspaces mathbb denote orthogonal projection mathbb following characterization purely unrectifiable sets holds mathcal m measurable subset mathbb mathcal infty purely m unrectifiable only there exists null subset universal bundle mid has mathcal n m mid replace mathcal m a