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We study the asymptotic behavior of three classes of nonlocal functionals incomplete metric spaces equipped with a doubling measure and supporting aPoincar\'e inequality. We show that the limits of these nonlocal functionalsare comparable to the variation $\| Df\|(\Omega)$ or the Sobolev semi-norm$\int_\Omega g_f^p\, d\mu$, which extends Euclidean results to metric measurespaces. In contrast to the classical setting, we also give an example to showthat the limits are not always equal to the corresponding total variation evenfor Lipschitz functions.

BV functions and nonlocal functionals in metric measure spaces