  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 by
smooth open sets. In the second part we prove some elementary properties of
local and global $s$-minimal sets, such as existence and compactness. We also
compare the two notions of minimizer (i.e. local and global), showing that in
bounded open sets with Lipschitz boundary they coincide. However, in general
this is not true in unbounded open sets, where a global $s$-minimal set may
fail to exist (we provide an example in the case of a cylinder
$\Omega\times\mathbb{R}$).
