Caccioppoli set

Remarks

Example: the unit ball $B \subseteq \R^n$ is a Caccioppoli set in $\R^n$. Its boundary is smooth, so $P(B) = \mathcal{H}^{n-1}(\partial B)$, the area of the unit sphere, which is finite.

Non-example: the half-space $H = \set{x \in \R^n : x_n > 0}$ with $n \ge 2$ has $P(H) = \infty$, since its boundary is a whole hyperplane. It still has locally finite perimeter, since $P(H, U)$ is the area of the part of the hyperplane inside $U$.

Equivalently, $E$ has finite perimeter in $\Omega$ exactly when the indicator function $\chi_E$ is a function of bounded variation in $\Omega$. Then the distributional gradient $D \chi_E$ is a vector-valued Radon measure, and $P(E, \Omega) = \abs{D\chi_E}(\Omega)$.