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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\Omega \\subseteq \\R^n$ be an open set.\nA [[Lebesgue measurable|2Z2QVWZ262]] set $E \\subseteq \\R^n$ is a *Caccioppoli set*, or a *set of finite perimeter*, in $\\Omega$ if its [[perimeter|Y68BHXZJBV]] $P(E, \\Omega)$ is finite.\nThe set $E$ has *locally finite perimeter* in $\\Omega$ if $P(E, U) \u003c \\infty$ for every open set $U$ whose closure is a compact subset of $\\Omega$.\n\n## Remarks\n\nExample: the unit ball $B \\subseteq \\R^n$ is a Caccioppoli set in $\\R^n$.\nIts boundary is smooth, so $P(B) = \\mathcal{H}^{n-1}(\\partial B)$, the area of the unit sphere, which is finite.\n\nNon-example: the half-space $H = \\set{x \\in \\R^n : x_n \u003e 0}$ with $n \\ge 2$ has $P(H) = \\infty$, since its boundary is a whole hyperplane.\nIt still has locally finite perimeter, since $P(H, U)$ is the area of the part of the hyperplane inside $U$.\n\nEquivalently, $E$ has finite perimeter in $\\Omega$ exactly when the indicator function $\\chi_E$ is a function of [[bounded variation|9YQ999QHHT]] in $\\Omega$.\nThen the distributional gradient $D \\chi_E$ is a vector-valued [[Radon measure|HZNEQQBCW4]], and $P(E, \\Omega) = \\abs{D\\chi_E}(\\Omega)$.\n",
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