$\Gamma$-convergence of non-local, non-convex functionals in one dimension

We study the $\Gamma$-convergence of a family of non-local, non-convex functionals in $L^p(I)$ for $p \ge 1$, where $I$ is an open interval. We show that the limit is a multiple of the $W^{1, p}(I)$ semi-norm to the power $p$ when $p>1$ (resp. the $BV(I)$ semi-norm when $p=1$). In dimension one, this extends earlier results which required a monotonicity condition.