Some remarks on Rademacher's theorem in infinite dimensions

Abstract

We provide an infinite dimensional version of Rademacher's theorem in a linear space provided with a bounded Radon measure μ. The underlying concepts of the Lipschitz property and differentiability hold μ-almost everywhere and only in the linear subspace of directions along which μ is quasiinvariant. The particular case where (X, μ) is the Wiener space (and for which the subspace of quasiinvariance coincides with the Cameron-Martin space) was proved in Enchev and Stroock (1993).