 The topological boundary formula_65 turns out to be too crude for Caccioppoli sets because its Hausdorff measure overcompensates for the perimeter formula_93 defined above.Indeed, the Caccioppoli set

representing a square together with a line segment sticking out on the left has perimeter formula_95, i.e.the extraneous line segment is ignored, while its topological boundary

has one-dimensional Hausdorff measure formula_97.