Asymptotics of the s-perimeter as s ↘ 0

We deal with the asymptotic behavior of the s-perimeter of a set E inside a domain Omega as s SE arrow 0. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of E and Omega. Moreover, we construct examples of sets for which the limit does not exist.