The Lip-lip condition on metric measure spaces

On complete metric spaces that support doubling measures, we show that the validity of a Rademacher theorem for Lipschitz functions can be characterised by Keith's "Lip-lip" condition. Roughly speaking, this means that at almost every point, the infinitesmal behavior of every Lipschitz function is essentially independent of the scales used in the blow-up at that point. Moreover, the doubling property can be further weakened to a local hypothesis on the measure; we also present results in this direction. Our techniques of proof are new and may be of independent interest. They include an explicit use of coordinate charts for measurable differentiable structures, as well as a blow-up procedure on Euclidean spaces that preserves Weaver derivations.