The main result of this paper is the following: any weighted Riemannian manifold
$(M,g,\unicode[STIX]{x1D707})$
, i.e., a Riemannian manifold
$(M,g)$
endowed with a generic non-negative Radon measure
$\unicode[STIX]{x1D707}$
, is infinitesimally Hilbertian, which means that its associated Sobolev space
$W^{1,2}(M,g,\unicode[STIX]{x1D707})$
is a Hilbert space.
We actually prove a stronger result: the abstract tangent module (à la Gigli) associated with any weighted reversible Finsler manifold
$(M,F,\unicode[STIX]{x1D707})$
can be isometrically embedded into the space of all measurable sections of the tangent bundle of
$M$
that are
$2$
-integrable with respect to
$\unicode[STIX]{x1D707}$
.
By following the same approach, we also prove that all weighted (sub-Riemannian) Carnot groups are infinitesimally Hilbertian.