  The worst case integration error in reproducing kernel Hilbert spaces of
standard Monte Carlo methods with n random points decays as $n^{-1/2}$.
However, re-weighting of random points can sometimes be used to improve the
convergence order. This paper contributes general theoretical results for
Sobolev spaces on closed Riemannian manifolds, where we verify that such
re-weighting yields optimal approximation rates up to a logarithmic factor. We
also provide numerical experiments matching the theoretical results for some
Sobolev spaces on the unit sphere and on the Grassmannian manifold. Our
theoretical findings also cover function spaces on more general sets such as
the unit ball, the cube, and the simplex.
