I am interested in the regularity of the "$n-1$ dimensional boundary" of the intersection of two Lipschitz boundaries, in particular I would like to know if this boundary always has zero $n-1$ dimensional measure.

For $i = 1, 2$ let $\Omega_i \subset \mathbb{R}^n$ open and disjoint with Lipschitz boundaries, i.e. for $x \in \partial \Omega_i$ there is a bi-Lipschitz map
$$
\Phi_{i, x}: U(x) \to B
$$
where $U(x)$ is an open neighbourhood of $x$, $B$ is the unit ball with center $0$ and $\Phi_{i, x}$ maps $U(x) \cap \Omega_i$ to the lower half space and $U(x) \cap \partial \Omega_i$ to the hyperplane $\{x^n = 0\}$. 
The case of interest is $\partial \Omega_1 \cap \partial \Omega_2$ non empty. Consider the sets
$$
F_i = \{ x \in \partial \Omega_i \mid \exists \epsilon > 0: B_\epsilon(x) \cap cl(\Omega_i)^c \subset \Omega_j \} \\
G_i = \{ x \in \partial \Omega_i \mid \exists \epsilon > 0: B_\epsilon(x) \cap cl(\Omega_i)^c \subset cl(\Omega_j)^c \} \\
N_i = \partial \Omega_i \setminus (F_i \cup G_i)
$$
where $j \neq i$. Now, intuitively $N_i$ is an $n-1$ dimensional boundary. Indeed, we see that
$$
F_i = \{ x \in \partial \Omega_i \cap \partial \Omega_j \mid \exists \epsilon > 0: B_\epsilon(x) \cap (\partial \Omega_i \cup \partial \Omega_j) \subset \partial \Omega_i \cap \Omega_j\}
$$
and $N_i = (\partial \Omega_i \cap \partial \Omega_j) \setminus F_i$. Thus $F_i$ is the interior of $\partial \Omega_i \cap \Omega_j$ in the subspace topology of $\partial \Omega_i \cup \partial \Omega_j$ and $N_i$ is the associated boundary. __Does $N_i$ have surface measure zero?__

In general boundaries do not have measure zero, however I wonder if the Lipschitz regularity of $\partial \Omega_i$ somehow induces enough regularity on the intersection $\partial \Omega_1 \cap \partial \Omega_2$ to imply this. I was trying to find a cone in the $\{x^n = 0\}$ plane such that the preimage under $\Phi_{i, x}$ is always contained either in $F_i$ or $G_i$, which would imply the desired result. If the result is not true in general, what other useful conditions are there besides the existence of such a cone?

Background: I consider broken Sobolev spaces, i.e. functions which are Sobolev functions in each of the $\Omega_i$ but not necessarily in the whole space. To examine the jump of their traces at the interface I would like to work locally as if the interface was the interface of two polygons (up to a set of zero surface measure). Usually this is considered for polygonal cells, for which you can find the desired cone (anyway, $N_i$ will be the union of $n-2$ dimensional affine spaces, hence it has $n-1$ dimensional measure zero) but I would like to know if this works in a more general setting.