  We establish functional analytic properties of the Stokes operator with
bounded measurable coefficients on $L^p_{\sigma} (\mathbb{R}^d)$, $d \geq 2$,
for $\lvert 1 / p - 1 / 2 \rvert < 1 / d$. These include optimal resolvent
bounds and the property of maximal $L^q$-regularity. We further give regularity
estimates on the gradient of the solution to the Stokes resolvent problem with
bounded measurable coefficients. As a key to these results we establish the
validity of a non-local Caccioppoli inequality to solutions of the Stokes
resolvent problem.
