  The $r$-parallel set to a set $A$ in Euclidean space consists of all points
with distance at most $r$ from $A$. Recently, the asymptotic behaviour of
volume and the surface area of parallel sets as $r$ tends to 0 has been studied
and some general results regarding their relations have been established. In
this paper we complete this picture. In particular, we show that a set is
Minkowski measurable if and only if it is S-measurable, i.e. if its S-content
is positive and finite, and that positivity and finiteness of lower and upper
Minkowski content implies the same for the S-contents and vice versa. The
results are formulated in the more general setting of Kneser functions.
Furthermore, the relations between Minkowski and S-contents are studied for
more general gauge functions. The results are also applied to simplify the
proof of the Modified Weyl-Berry conjecture in dimension one.
