  We investigate weighted Sobolev spaces on metric measure spaces $(X,d,m)$.
Denoting by $\rho$ the weight function, we compare the space $W^{1,p}(X,d,\rho
m)$ (which always concides with the closure $H^{1,p}(X,d,\rho m)$ of Lipschitz
functions) with the weighted Sobolev spaces $W^{1,p}_\rho(X,d,m)$ and
$H^{1,p}_\rho(X,d,m)$ defined as in the Euclidean theory of weighted Sobolev
spaces. Under mild assumptions on the metric measure structure and on the
weight we show that $W^{1,p}(X,d,\rho m)=H^{1,p}_\rho(X,d, m)$. We also adapt
results by Muckenhoupt and recent work by Zhikov to the metric measure setting,
considering appropriate conditions on $\rho$ that ensure the equality
$W^{1,p}_\rho(X,d,m)=H^{1,p}_\rho(X,d,m)$.
