Let be a monotone submodular set function, and let be a matroid. We consider the problem . It is known that the greedy algorithm yields a -approximation [M. L. Fisher, G. L. Nemhauser, and L. A. Wolsey, Math. Programming Stud., no. 8 (1978), pp. 73–87] for this problem. For certain special cases, e.g., , the greedy algorithm yields a -approximation. It is known that this is optimal both in the value oracle model (where the only access to f is through a black box returning for a given set S) [G. L. Nemhauser and L. A. Wolsey, Math. Oper. Res., 3 (1978), pp. 177–188] and for explicitly posed instances assuming [U. Feige, J. ACM, 45 (1998), pp. 634–652]. In this paper, we provide a randomized -approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [J. Combin. Optim., 8 (2004), pp. 307–328], and a continuous greedy process that may be of independent interest. As a special case, our algorithm implies an optimal approximation for the submodular welfare problem in the value oracle model [J. Vondrák, Proceedings of the th ACM Symposium on Theory of Computing, 2008, pp. 67–74]. As a second application, we show that the generalized assignment problem (GAP) is also a special case; although the reduction requires to be exponential in the original problem size, we are able to achieve a -approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constraints.