  We consider the Hausdorff dimension of random covering sets generated by
balls and general measures in Euclidean spaces. We prove, for a certain
parameter range, a conjecture by Ekstr\"om and Persson concerning the exact
value of the dimension in the special case of radii
$(n^{-\alpha})_{n=1}^\infty$. For generating balls with an arbitrary sequence
of radii, we find sharp bounds for the dimension and show that the natural
extension of the Ekstr\"om-Persson conjecture is not true in this case.
Finally, we construct examples demonstrating that there does not exist a
dimension formula involving only the lower and upper local dimensions of the
measure and a critical parameter determined by the sequence of radii.
