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Let $M^m$ be an oriented Lipschitz submanifold with boundary of $\mathbb{R}^n$. One can ask whether the boundary of $M$ in the sense of currents coincides with the topological boundary.

A celebrated result by Federer about finite perimeter sets implies that the answer is positive if $m=n$, while a paper of Harrison proves it in general codimension (Theorem 3.iii, https://arxiv.org/abs/math/9310231).

I am asking whether there exist a proof of the general codimension case that avoids the use of $(\lambda,n)$-sets introduced by Harrison and directly generalize the methods of Federer?

More specifically, I want to know if there is a way to define a notion of measure theoretic boundary in arbitrary codimension that allows one to prove something along the lines of "if the measure theoretic boundary of a countably rectificable set has finite Hausdorff measure, then the measure theoretic boundary and the boundary in the sense of currents coincide"?

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