  We consider a Kantorovich potential associated to an optimal transportation
problem between measures that are not necessarily absolutely continuous with
respect to the Lebesgue measure, but are comparable to the Lebesgue measure
when restricted to balls with radius greater than some $\delta>0$. Our main
results extend the classical regularity theory of optimal transportation to
this framework. In particular, we establish both H\"older and Sobolev
regularity results for Kantorovich potentials up to some critical length scale
depending on $\delta$. Our assumptions are very natural in the context of the
numerical computation of optimal maps, which often involves approximating by
sums of Dirac masses some measures that are absolutely continuous with
densities bounded away from zero and infinity on their supports.
