Lusin-type theorems for Cheeger derivatives on metric measure spaces

TL;DR

This paper extends Lusin-type theorems to metric measure spaces with doubling measures and Poincaré inequalities, demonstrating that Borel functions can be approximated by derivatives of continuous functions in these spaces.

Contribution

It establishes Lusin-type theorems for Cheeger derivatives on metric measure spaces, generalizing classical results from Euclidean spaces.

Findings

Lusin-type theorems hold for Cheeger derivatives in metric measure spaces

Borel functions are almost everywhere derivatives of continuous functions in these spaces

Results apply to spaces with doubling measures and Poincaré inequalities

Abstract

A theorem of Lusin states that every Borel function on $R$ is equal almost everywhere to the derivative of a continuous function. This result was later generalized to $R^n$ in works of Alberti and Moonens-Pfeffer. In this note, we prove direct analogs of these results on a large class of metric measure spaces, those with doubling measures and Poincar\'e inequalities, which admit a form of differentiation by a famous theorem of Cheeger.