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

Silvano Delladio

Annales Polonici Mathematici, Tome 101 (2011), p. 73-78 / Harvested from The Polish Digital Mathematics Library

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 ℝⁿ. For V ∈ G(n,m) let $π_V$ denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an $ ^m$-measurable subset of ℝⁿ with $ ^m(A) < ∞$. Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle ${(V,v)| V ∈ G(n,m), v ∈ V}$ such that, for all P ∈ A, one has $ ^{m(n-m)}({V ∈ G(n,m)|(V,π_V(P)) ∈ Z}) > 0$. One can replace “for all P ∈ A” by “for $ ^m$-a.e. P ∈ A”.

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Publié le : 2011-01-01

Zbl 1234.28007

EUDML-ID : urn:eudml:doc:280880