We say that a $\mathscr{H}^{m}$-measurable set $M\subset\mathbb{R}^n$ is $m$-rectifiable if there are countably many Lipschitz maps $f_j:\mathbb{R}^m\to \mathbb{R}^n$ and a $\mathscr{H}^m$-null set $M_0$ such that $M\subset M_0\bigcup f_j(\mathbb{R}^m)$.

Let $M\subset\mathbb{R}^n$ be a $\mathscr{H}^{m}$-locally finite $m$-rectifiable set. Define the $m$-dimensional measure-theoretic interior, exterior and boundary of $M$ as follows. $$ \begin{aligned} \text{Int}( M):=\{x\in\mathbb{R}^n \,|\,\lim_{r\to 0}\frac{\mathscr{H}^m(M\cap B_r(x))}{\omega_m r^m}=1\},\\ \text{Ext} (M):=\{x\in\mathbb{R}^n \,|\,\lim_{r\to 0}\frac{\mathscr{H}^m(M\cap B_r(x))}{\omega_m r^m}=0\},\\ \partial^* M:=\mathbb{R}^n\setminus (\text{Int}(M)\cup \text{Ext}(M)).\\ \end{aligned} $$

It can be proven that $\mathscr{H}^m(M\Delta \text{Int}(M))=0$ and that $\mathscr{H}^m((\mathbb{R}^n\setminus M)\Delta \text{Ext}(M))=0$ ($\Delta$ denotes the simmetric difference).

Now endow $M$ with a measurable orientation, so that $M$ becomes an oriented, $\mathscr{H}^{m}$-locally finite $m$-rectifiable set in $\mathbb{R}^n$. Denote with $[[M]]$ the current canonically associated to $M$. Note that then finite perimeter sets are just as the (obviously $\mathscr{H}^n$-locally finite and trivially oriented) $n$-rectifiable sets $E\subset\mathbb{R}^n$ such that $\mathbb{M}(\partial[[E]])<\infty$, where $\partial$ and $\mathbb{M}$ are respectively the boundary operator and the mass operator for currents. The following two results are well known.

(1) De Giorgi's Structure Theorem: If a measurable set $E\subset\mathbb{R}^n$ has finite perimeter, then $\partial^*E$ is an oriented $(n-1)$-rectifiable set such that $\mathscr{H}^{n-1}(\partial^*E)<\infty$ and $\partial [[E]]=[[\partial^* E]]$.

(2) Federer's Characterization Theorem: If a measurable set $E\subset \mathbb{R}^n$ is such that $\mathscr{H}^{n-1}(\partial^*E)<\infty$, then $E$ has finite perimeter.

It is natural to ask if $(1)$ and $(2)$ can be generalized to hold for oriented, $\mathscr{H}^{m}$-locally finite $m$-rectifiable sets $M\subset\mathbb{R}^n$. In fact, Federer and Fleming proved that De Giorgi's theorem generalizes to the following.

(1)' Boundary Rectifiability Theorem: If an oriented, $\mathscr{H}^{m}$-locally finite $m$-rectifiable set $M\subset\mathbb{R}^n$ satisfies $\mathbb{M}(\partial[[M]])<\infty$, then there exists an $(m-1)$-dimensional integer-multiplicity current $S=\langle N,\theta,\xi\rangle$ such that $\partial [[M]]=S$.

The notation $S=\langle N,\theta,\xi\rangle$ means that $S$ acts on test forms as the measure $\mathscr{H}^{m-1}_{|N}(dx)\theta(x)\xi(x)$, where $N$ is a $(m-1)$-rectifiable set, $\theta$ a measurable integer-value function positive and $\mathscr{H}^{m-1}$-locally integrable on $N$, and for $\mathscr{H}^{m-1}$-a.e. $x$ $\xi(x)$ is a $(m-1)$-multivector of norm $1$ orienting $T_xN$. An important observation is that the multiplicity $\theta$ must be integer-valued but might be different from $1$, so in general $S$ cannot be identified with a set.

I have two questions.

First question: Is it true that $\mathscr{H}^{m-1}(N\Delta \partial^*M))=0$ (i.e. that we can take $N=\partial^*M$ in the theorem)?

Second question: Is something known about the generalization of Federer's theorem? I would conjecture the following.

(2)' Conjecture. If an oriented, $\mathscr{H}^{m}$-locally finite $m$-rectifiable set $M\subset\mathbb{R}^n$ is such that $\mathscr{H}^{m-1}(\partial^*M)<\infty$, then $\mathbb{M}(\partial [[M]])<\infty$. More generally, if $T=\langle M,\theta,\xi\rangle$ is an $m$-dimensional integer-multiplicity current such that $\mathscr{H}^{m-1}(\partial^* M)<\infty$, then $\mathbb{M}(\partial T)<\infty$.

  • 1
    $\begingroup$ What is a boundary in the sense of currents? (The linked paper by Harrison does not refer to currents.) And what would qualify a notion of boundary as measure-theoretic — what can it depend on or what transformations should preserve it? $\endgroup$
    – Matt F.
    Jan 20, 2022 at 16:43
  • 3
    $\begingroup$ If $T$ is a current that acts on compactly supported $m$-forms of $\mathbb{R}^n$, its boundary $\partial T$ is defined to be the current that acts on $k-1$-forms as $<\partial T, \omega >=<T,d\omega>$. If $M^m$ is a smooth submanifold of $\mathbb{R}^n$, can think of $M$ as a current $T_M$ via $<T_M,\omega>=\int_M \omega$. Stokes theorem says that $\partial T_M=T_{\partial M}$ (ie boundary in the sense of currents and topological boundary coincide). As a corollary of the result of Harrison, we know that this continues to hold when $M$ is assumed only Lipschitz. $\endgroup$
    – Titti
    Jan 21, 2022 at 13:36
  • $\begingroup$ I think notation needed in the body of the question should be in the question, not the comments. $\endgroup$
    – LSpice
    Jan 31, 2022 at 21:10
  • 1
    $\begingroup$ Please be aware that every edit of a question or of one of its answers bumps the thread to the front page. This has happened for this thread already more than 20 times, and this is a nuisance for other users. Please refrain from unnecessary edits to your posts. -- Usually, the vast majority of minor edits can be avoided by writing and proofreading a question or an answer (or any update to such) carefully before posting it. $\endgroup$
    – Stefan Kohl
    Sep 16, 2022 at 17:03


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