  The main result of this paper is the following: any `weighted' Riemannian
manifold $(M,g,\mu)$ - i.e. endowed with a generic non-negative Radon measure
$\mu$ - is `infinitesimally Hilbertian', which means that its associated
Sobolev space $W^{1,2}(M,g,\mu)$ is a Hilbert space.
  We actually prove a stronger result: the abstract tangent module (\`a la
Gigli) associated to any weighted reversible Finsler manifold $(M,F,\mu)$ can
be isometrically embedded into the space of all measurable sections of the
tangent bundle of $M$ that are $2$-integrable with respect to $\mu$.
