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In the first part of this paper we show that a set $E$ has locally finite$s$-perimeter if and only if it can be approximated in an appropriate sense bysmooth open sets. In the second part we prove some elementary properties oflocal and global $s$-minimal sets, such as existence and compactness. We alsocompare the two notions of minimizer (i.e. local and global), showing that inbounded open sets with Lipschitz boundary they coincide. However, in generalthis is not true in unbounded open sets, where a global $s$-minimal set mayfail to exist (we provide an example in the case of a cylinder$\Omega\times\mathbb{R}$).

Approximation of sets of finite fractional perimeter by smooth sets and comparison of local and global $s$-minimal surfaces