  We consider a generalized steady Stokes system with discontinuous
coefficients in a nonsmooth domain when the inhomogeneous term belongs to a
weighted $L^q$ space for $2<q<\infty$. We prove the global weighted
$L^q$-estimates for the gradient of the weak solution and an associated
pressure under the assumptions that the coefficients have small BMO (bounded
mean oscillation) semi-norms and the domain is sufficiently flat in the
Reifenberg sense. On the other hand, a given weight is assumed to belong to a
Muckenhoupt class. Our result generalizes the global $W^{1.q}$ estimate of
Calder\'{o}n-Zygmund with respect to the Lebesgue measure for the Stokes system
in a Lipschitz domain.
