Federer's characterization of sets of finite perimeter states (in Euclidean spaces) that a set is of finite perimeter if and only if the measure-theoretic boundary of the set has finite Hausdorff measure of codimension one. In complete metric spaces that are equipped with a doubling measure and support a Poincar\'e inequality, the "only if" direction was shown by Ambrosio (2002). By applying fine potential theory in the case $p=1$, we prove that the "if" direction holds as well.

中文翻译：

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有限周长集的新费德勒型表征

Federer 对有限周长状态集（在欧几里德空间中）的表征，即当且仅当该集合的测度理论边界具有有限的 Hausdorff 测度的 codimension 1 时，该集合是有限周长的。在配备加倍测度并支持 Poincar\'e 不等式的完全度量空间中，Ambrosio (2002) 显示了“仅当”方向。通过在 $p=1$ 的情况下应用精细势理论，我们证明“if”方向也成立。