Geometry of Euclidean Measures: Densities and Rectifibility
Abstract
The aim of these lectures is to prove the following fundamental theorem in Geometric Measure Theory, due to D. Preiss (Ann. of Math. (2) 125 (1987), 537–643).
Let μ be a locally finite measure on Rn and k ≥ 0 a real number. Assume that the limit
exists, is finite and nonzero for μ-a.e. x. Then either μ = 0 or k ≤ n is integer. Moreover, in the latter case μ is a k-rectifiable measure, i.e. there exist a measurable function f and a countable familyΓi of k-dimensional Lipschitz submanifolds such that
, for every Borel set A. Here Hk denotes the natural k-dimensional volume measure.
Keywords
Geometric Measure Theory, Rectifiable Sets, Tangent Measures, Preiss Theorem Prerequisites: Basic measure theory, Lipschitz functions, Hausdorff measures
Audience
MSc students, PhD students, Postdocs
Language
English
Remarks and notes
The main tools and prerequisites will be quickly recalled, without proofs.