A Note on Alberti's Luzin-Type Theorem for Gradients

TL;DR

This paper provides an elementary geometric measure theory and topology-based proof of Alberti's Luzin-type theorem for gradients, with applications to $C^2$-rectifiability, simplifying previous approaches.

Contribution

It offers a new, simplified proof of Alberti's Luzin-type theorem for gradients using elementary methods, and discusses implications for rectifiability.

Findings

Simplified proof of Alberti's Luzin-type theorem

Applications to $C^2$-rectifiability problem

Enhanced understanding of gradient measure properties

Abstract

We give a "soft" proof of Alberti's Luzin-type theorem in [1] (G. Alberti, A Lusintype theorem for gradients, J. Funct. Anal. 100 (1991)), using elementary geometric measure theory and topology. Applications to the $C^2$-rectifiability problem are also discussed.