In the article "Funzioni BV e tracce" by Anzellotti and Giaquinta (MR555952, Zbl 0432.46031), at page 6 you can read (assume $\Omega \subset \mathbb{R}^n$ open): "The following example shows that the hypothesis $\mathcal{H}^{n-1} (\partial \Omega) < + \infty$ cannot be replaced by the hypothesis $P(\Omega)< + \infty$, even if the the topological frontier and the reduced boundary of $\Omega$ coincide."
Now, I know that in general topological boundary and reduced boundary have nothing to do with each other, but I thought that if you have a set of finite perimeter, by De Giorgi's structure theorem they were essentially the same, in the sense that you can work equivalently with the perimeter or the Hausdorff measure (with the reduced boundary). Namely if you have a set of finite perimeter $E \subset \mathbb{R}^n$, by the structure theorem:
$$ D 1_E = \nu_E \mathcal{H}^{n-1} |_{\partial^* \Omega} ,$$
but if $\partial E= \partial^* E$, then:
$$+ \infty > P(E) = |D1_E|(\mathbb{R}^n)= \mathcal{H}^{n-1} (\partial ^* E)= \mathcal{H}^{n-1} (\partial E).$$
What's the caveat here that I am not seeing?
