A discrete coarea-type formula for the Mumford-Shah functional in dimension one

We consider the convex lift of Mumford-Shah type functionals in the space of rectifiable currents and we prove a generalized coarea formula in dimension one, for finite linear combinations of SBV graphs. Then we use this result to prove the equivalence between the minimum problems for the Mumford-Shah functional and the lifted one and, as a consequence, we obtain a weak existence result for calibrations in one space dimension.