Generalized rectifiability of measures and the identification problem

Abstract

One goal of geometric measure theory is to understand how measures in the plane or a higher dimensional Euclidean space interact with families of lower dimensional sets. An important dichotomy arises between the class of rectifiable measures, which give full measure to a countable union of the lower dimensional sets, and the class of purely unrectifiable measures, which assign measure zero to each distinguished set. There are several commonly used definitions of rectifiable and purely unrectifiable measures in the literature (using different families of lower dimensional sets such as Lipschitz images of subspaces or Lipschitz graphs), but all of them can be encoded using the same framework. In this paper, we describe a framework for generalized rectifiability, review a selection of classical results on rectifiable measures in this context, and survey recent advances on the identification problem for Radon measures that are carried by Lipschitz or Hölder or \(C^{1,\alpha }\) images of Euclidean subspaces, including theorems of Azzam–Tolsa, Badger–Schul, Badger–Vellis, Edelen–Naber–Valtorta, Ghinassi, and Tolsa–Toro.

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  • 24 April 2019

    The article, “Generalized rectifiability of measures and the identification problem ”, written by Matthew Badger was originally published electronically on the publisher’s internet portal (currently SpringerLink) on March 6, 2019, as open access, with “© The Author(s)”; instead, it should be “© Springer Nature Switzerland AG” and the article is forthwith distributed under the terms of copyright.

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The author was partially supported by NSF DMS Grants 1500382 and 1650546.

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This survey paper is based on a talk at the Northeast Analysis Network Conference held in Syracuse, New York in September 2017.

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Badger, M. Generalized rectifiability of measures and the identification problem. Complex Anal Synerg 5, 2 (2019). https://doi.org/10.1007/s40627-019-0027-3

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Keywords

  • Structure of measures
  • Atoms
  • Generalized rectifiability
  • Fractional rectifiability
  • Density ratios
  • Flatness
  • Geometric square functions

Mathematics Subject Classification

  • Primary 28A75
  • Secondary 26A16
  • 42B99
  • 54F50