A convex sweeping process is considered in a separable Hilbert space. The majority of works on sweeping processes use the Hausdorff distance to describe the movement of the convex set generating the process. However, for unbounded sets the use of the Hausdorff distance does not always guarantee the fulfilment of conditions under which a solution exists. In the present work, instead of the Hausdorff distance we use the.excesses of sets. These excesses are subjected to positive Radon measures depending on.. We prove the existence of right continuous BV solutions and establish their dependence on single-valued perturbations depending only on time. The results we obtain are applied to prove the theorems on existence and relaxation of extremal right continuous BV solutions of a sweeping process with multivalued perturbations. Some results on absolutely continuous solutions are derived as corollaries.