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

Silvano Delladio. "A criterion for pure unrectifiability of sets (via universal vector bundle)." Annales Polonici Mathematici 102.1 (2011): 73-78. <http://eudml.org/doc/280880>.

@article{SilvanoDelladio2011,
abstract = {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”.},
author = {Silvano Delladio},
journal = {Annales Polonici Mathematici},
keywords = {purely unrectifiable sets; rectifiable sets; geometric measure theory},
language = {eng},
number = {1},
pages = {73-78},
title = {A criterion for pure unrectifiability of sets (via universal vector bundle)},
url = {http://eudml.org/doc/280880},
volume = {102},
year = {2011},
}

TY - JOUR
AU - Silvano Delladio
TI - A criterion for pure unrectifiability of sets (via universal vector bundle)
JO - Annales Polonici Mathematici
PY - 2011
VL - 102
IS - 1
SP - 73
EP - 78
AB - 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”.
LA - eng
KW - purely unrectifiable sets; rectifiable sets; geometric measure theory
UR - http://eudml.org/doc/280880
ER -