Lahti (2018): Federer's characterization of sets of finite perimeter

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theorem_1_1: Finite codimension-one boundary measure implies finite perimeter in the stated metric setting. In the Euclidean plane the criterion and the perimeter representation imply that a measurable set with empty measure-theoretic boundary is null or conull.

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Panu Lahti, Federer's characterization of sets of finite perimeter in metric spaces, arXiv:1804.11216v1 (27 April 2018). The title page is dated 1 May 2018. The version used here is the arXiv v1 PDF; its page numbers and labels are used below.

The paper proves the finite-perimeter criterion in complete metric spaces with a doubling measure and a $(1,1)$-Poincare inequality. If the codimension-one Hausdorff measure of a measurable set's measure-theoretic boundary in an open set is finite, then its perimeter in that open set is finite (Theorem 1.1, p. 2). The Euclidean case, due to Federer, is recalled on p. 1 with the reference Geometric Measure Theory (1969), Section 4.5.11. The metric proof uses fine potential theory for $p=1$; that proof is not consumed here.

The criterion and its Euclidean application record the boundary definition (2.3), the perimeter representation (2.4), and the relative isoperimetric inequality (2.2). They give an explicit local-to-global deduction: a measurable subset of the plane with empty measure-theoretic boundary is null or conull. Empty topological boundary is not the hypothesis, and no regularity of the coloring is assumed beyond measurability.

Source and terms. arXiv:1804.11216v1. The arXiv record gives the non-exclusive distribution license, read on 2026-10-17; all other rights are reserved. No file of this source is held.

Reading. The introduction's Euclidean criterion, Theorem 1.1, the boundary definition and formulas (2.2)--(2.4), and the standing metric hypotheses were checked at statement level in the v1 TeX and PDF pages 1--3 and 5--6. The corresponding Euclidean specialization is recorded on the result page. Neither Federer's original proof nor Lahti's proof of the metric extension was read. No independent proof verification is asserted.

Bears on

• Problem 173: supplies the empty-boundary implication for the measurable two-scale approach, whose geometry must first establish that the measure-theoretic boundary is empty. The source proves no triangle-coloring theorem.