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

作者：Delladio Silvano^*

来源：Annales Polonici Mathematici, 2011, 102(1): 73-78.

DOI：10.4064/ap102-1-6

• purely unrectifiable sets
• rectifiable sets
• geometric measure theory

Let m,91 be positive integers such that m < n and let G(n, m) be the Grassmann manifold of all m-dimensional subspaces of R(n). For V is an element of G(n, m) let pi(V) denote the orthogonal projection from R(n) onto V. The following characterization of purely unrectifiable sets holds. Let A be an H(m)-measurable subset of R(n) with H(m)(A) < infinity. Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle {(V, v) vertical bar V is an element of G(n,m), v is an element of V} such that, for all P is an element of A, one has H(m(n-m))({V is an element of G(n,m) vertical bar (V, pi(V). (P)) is an element of Z}) > 0. One can replace "for all P is an element of A" by "for H(m)-a.e. P is an element of A".

• 出版日期2011