A finite perimeter set is a Lebesgue measurable set 
whose perimeter
 |
is finite. The supremum is over continuously differentiable vector fields 
with compact
support and 
for every 
,
where the norm is Euclidean and 
denotes divergence. The set
need not have finite Lebesgue measure (Maggi
2012).
Changing 
on a set of Lebesgue measure zero leaves 
unchanged. For a bounded open
set with smooth boundary, 
equals the 
-dimensional Hausdorff
measure of that boundary, recovering the usual boundary surface area.
On a Riemannian manifold, the analogous definition
uses tangent vector fields, the metric norm
and divergence, and Riemannian volume.
This notion permits the isoperimetric inequality
to be formulated without requiring a smooth boundary.
