Under minimal conditions precise bounds are obtained for the expectation of the supremum of the weighted empirical process over the interval $(0, 1/(n(\log n)^{d - 1}))$, where $d$ is the dimension of the underlying random vectors. The allowed growth of the weight function is optimal in the iid case. The results will have broad applications in the theory of all kinds of nonstandard weighted empirical processes, such as empirical processes based on uniform spacings or $U$-statistics, where it is often not so easy to show directly (as in the iid case) that the considered suprema converge to 0 in probability.