Lusin-type theorems for Cheeger derivatives on metric measure spaces

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.