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  "body": "\n## Summary\n\nA Caccioppoli set, or set of finite perimeter, is a measurable subset of $\\R^n$ whose [[perimeter|Y68BHXZJBV]] is finite.\nEquivalently, its indicator function has [[bounded variation|9YQ999QHHT]].\nThese sets are the measure-theoretic replacement for regions with smooth boundary.\nThey are closed under the limits that appear when one minimizes area.\nSo they give a setting where problems such as the isoperimetric problem have solutions.\n\n## Explanations\n\n### L1 human\n\nMany regions that come up as limits of nice regions have wild boundaries, and the classical notion of surface area breaks down for them.\nA Caccioppoli set is a region whose boundary has finite area in a weak sense that still works for such limits.\nAny bounded region in the plane with a polygonal boundary is a Caccioppoli set.\nThe union of the disjoint discs of radius $2^{-k}$ centred at $(3k, 0)$, for $k = 1, 2, \\dots$, is also one, since its perimeter is $\\sum_k 2\\pi 2^{-k} = 2\\pi$.\nThe union of the discs of radius $1$ centred at the same points is not, since its perimeter is infinite.\n",
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