A note on a residual subset of Lipschitz functions on metric spaces

TL;DR

This paper proves that in certain metric spaces, the set of Lipschitz functions with non-zero point-wise Lipschitz constant almost everywhere is residual, extending a known result from Euclidean spaces to more general metric spaces.

Contribution

It generalizes a Euclidean space result to quasi-convex, complete, separable metric spaces, showing the residuality of Lipschitz functions with non-zero point-wise Lipschitz constants.

Findings

Residual set of Lipschitz functions with non-zero point-wise Lipschitz constant is dense

Extension of Euclidean space results to general metric spaces

Provides a metric space analogue of a known Euclidean result

Abstract

Let (X, d) be a quasi-convex, complete and separable metric space with reference probability measure m. We prove that the set of of real valued Lipschitz function with non zero point-wise Lipschitz constant m-almost everywhere is residual, and hence dense, in the Banach space of Lipschitz and bounded functions. The result is the metric analogous of a result proved for real valued Lipschitz maps defined on R2 by Alberti, Bianchini and Crippa in [1].