Geometry of Euclidean Measures: Densities and Rectifibility

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 familyΓ_i of k-dimensional Lipschitz submanifolds such that

, for every Borel set A. Here H^k 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.