Lusin’s theorem for derivatives with respect to a continuous function

Abstract

For a nowhere constant continuous function g g on a real interval I I and for a Borel measure μ \mu on I I , we give simple necessary and sufficient conditions guaranteeing, for any Borel function f f on I I , the existence of a continuous function F F on I I such that the derivative of F F with respect to g g is, μ \mu almost everywhere, equal to f f .