Given two measures \(\mu ,\nu \) on \(\mathbb {R}^d\) that satisfy Carleman’s condition, we provide a numerical scheme to approximate as closely as desired the total variation distance between \(\mu \) and \(\nu \) . (In particular, the supports of \(\mu \) and \(\nu \) are not necessarily compact.) It consists of solving a sequence (hierarchy) of convex relaxations whose associated sequence of optimal values converges to the total variation distance, an additional illustration of the versatility of the Moment-SOS hierarchy. Each relaxation in the hierarchy is a semidefinite program whose size increases with the number of involved moments. It has an optimal solution which is a couple of degree-2n pseudo-moments which converge, as n grows, to moments of the Hahn-Jordan decomposition of \(\mu -\nu \) . Illustrative examples are provided.
