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

Annales Polonici Mathematici (2011)

- Volume: 102, Issue: 1, page 73-78
- ISSN: 0066-2216

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topSilvano 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 -

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