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Source. P. Lahti, Federer's characterization of sets of finite perimeter in metric spaces, arXiv:1804.11216v1, Theorem 1.1 (p. 2), with (2.2) and (2.3) on p. 5 and (2.4) on p. 6, under the standing hypotheses in Section 2 (p. 3). The Euclidean criterion recalled on p. 1 is attributed to H. Federer, Geometric Measure Theory (1969), Section 4.5.11. The source card identifies the version and reading limits.

Read depth. Claims checked: the stated hypotheses, boundary definition and three formulas were read in the v1 TeX and on the PDF page images. The original proofs of the criterion and representation were not read. The application below is a direct author-recorded deduction by Plasma AI.

Statement and convention

Let $(X,d)$ be a complete metric space with at least two points and let $\mu$ be a Borel regular outer measure, positive and finite on balls and satisfying the doubling condition. Suppose that $(X,d,\mu)$ supports a $(1,1)$-Poincare inequality. Write $P(A,\Omega)$ for the variation of the indicator of a measurable set $A$ in the open set $\Omega$, and $\mathcal H$ for the codimension-one Hausdorff measure formed from ball weights $\mu(B(x,r))/r$. The measure-theoretic boundary is

$$\partial^* A= \left\{x: \limsup_{r\downarrow0}\frac{\mu(A\cap B(x,r))}{\mu(B(x,r))}>0, \quad \limsup_{r\downarrow0}\frac{\mu(B(x,r)\setminus A)}{\mu(B(x,r))}>0 \right\}.$$

Theorem 1.1. If $\mathcal H(\partial^*A\cap\Omega)<\infty$, then $P(A,\Omega)<\infty$.

For finite-perimeter sets the accompanying representation (2.4) gives

$$P(A,U)=\int_{\partial^*A\cap U}\theta_A\,d\mathcal H$$

for Borel $U\subset\Omega$, with $\theta_A$ bounded above and below by positive constants depending on the metric-space constants. The relative isoperimetric inequality (2.2) gives

$$\min\{\mu(A\cap B(x,r)),\mu(B(x,r)\setminus A)\} \le 2C_P r\,P(A,B(x,\lambda r)),$$

where $C_P$ and $\lambda$ are Poincare constants. These formulas retain the source's normalization; no exact normalization of $\mathcal H$ is needed when the boundary is empty.

Euclidean application

Let $A\subset\mathbb R^2$ be Lebesgue measurable and suppose that $\partial^*A=\varnothing$. Then $A$ is null or conull.

Indeed, in every open ball $\Omega$ the criterion gives finite perimeter, and (2.4) gives $P(A,\Omega)=0$. Apply (2.2) to any ball, taking a larger open ball for its dilate if necessary. At least one of $A$ and its complement has zero measure in that ball. For the nested balls $B(0,n)$, $n\ge1$, the same choice must hold for every $n$: two different choices would contradict the positive measure of the smaller ball. Their union is the plane, proving the assertion.

For a measurable red-blue coloring let $A^*$ be the red density-one points and $B^*$ the red density-zero points. Then $\mathbb R^2\setminus(A^*\cup B^*)=\partial^*A$, exactly with the limsup convention above. Indeed, failure of both density limits $0$ and $1$ is equivalent to positive upper density of each class. Thus proving this complement empty suffices for the application; proving only that it has area zero does not.

The criterion is used locally. No finite area, boundedness, smoothness or polygonality of $A$ is required. The measure-theoretic boundary is not the topological boundary or a boundary defined only at density $1/2$.

Bears on

• Problem 173: supplies the last measure-theoretic implication for an approach that first forces the boundary of a measurable avoiding coloring to be empty. The geometric premise remains the responsibility of the two-scale argument.