Asymptotics of non-local perimeters

We introduce a notion of non-local perimeter which is defined through an arbitrary positive Borel measure on $${\mathbb {R}}^d$$Rd which integrates the function $$1\wedge |x|$$1∧|x| . Such definition of non-local perimeter encompasses a wide range of perimeters which have been already studied in the literature, including fractional perimeters and anisotropic fractional perimeters. The main part of the article is devoted to the study of the asymptotic behaviour of non-local perimeters. As direct applications we recover well-known convergence results for fractional perimeters and anisotropic fractional perimeters.