In this talk, regularization of ill-posed inverse problems involving Radon measures is studied. The main motivation is to nd a sparse reconstruction for an unknown Radon measure based on indirect measurements. Similar questions has been previously examined in [3]. In this talk, it is shown that the minimization of the Tikhonov functional with the total variation norm penalty is a well-posed problem for Radon measures. A minimizer of the Tikhonov functional is characterized by using the corresponding Fenchel predual problem. Convergence and convergence rate properties of minimizers are examined under common source conditions, like in [1, 2]. A numerical algorithm promoting sparsity is proposed. The work has been done in collaboration with Kristian Bredies, University of Graz, Austria.