  The goal of the paper is to prove the equivalence of distributional and
synthetic Ricci curvature lower bounds for a weighted Riemannian manifold with
continuous metric tensor having Christoffel symbols in $L^2_{{\rm loc}}$, and
with weight in $C^0\cap W^{1,2}_{{\rm loc}}$. The regularity assumptions are
sharp, in the sense that they are minimal in order to define the distributional
Ricci curvature tensor.
