TITLE: Possible application of divergence Theorem?
QUESTION [7 upvotes]: suppose that $f \in C^1 (\mathbb{R}^{N+1},\mathbb{R})$. It's well known that if all his points are regular points i.e.
$$\nabla f (x) \neq 0 \; \; \; \forall x \in \mathbb{R}^{N+1}$$
then, for every fixed $c \in \mathbb{R}$, the level set:
$$f^{-1}(c)=\{x \in \mathbb{R}^{N+1}: f(x)=c\}$$
is a $C^1$ manifold of dimension $N$.
So if $\Omega:=\left \{ x \in \mathbb{R}^{N+1}: f(x)< c \right \}$, supposed bounded, with enough regularity on a vector field $G$, I can apply the classical divergence theorem:
$$\int_{\Omega} \operatorname{div}(G) \,\mathrm{d}\mathcal{L}^{N+1}=\int_{\partial \Omega}\langle G,\nu\rangle \,\mathrm{d}H_N$$
where $H_N$ is the $N$-dimensional Hausdorff measure.
Now suppose that not all his points are regular and let $\operatorname{crit}(f)$ be the set of critical points of $f$ i.e.
$$\operatorname{crit}(f):=\{x \in \mathbb{R}^{N+1}: \nabla f (x)=0 \}$$
Now, suppose that for a fixed $\bar{c} \in \mathbb{R}$, I can prove that:
$$H_N \left( f^{-1}(\bar{c}) \cap \operatorname{crit}(f) \right)=0 $$
So $f^{-1}(\bar{c}) \setminus \operatorname{crit}(f)$ remains a $C^1$ manifold of dimension $N$, while $f^{-1}(\bar{c}) \cap \operatorname{crit}(f)$ is set of measure zero. So the points of $f^{-1}(\bar{c})$ have $\nabla f(x) \neq 0$, except for a set of measure zero.
My question is: is possible to apply divergence theorem to $\Omega=\left \{ x \in \mathbb{R}^{N+1}: f(x)< \bar{c} \right \}$ also in this case?
I found a lot of generalization of divergence theorem but apparently none for my case. 
Thanks.

REPLY [3 votes]: In your case the boundary has (locally) finite $H^N$ measure and such sets have finite perimeter. It is known that for sets of finite perimeter the divergence theorem is true. Later I will write a more complete answer, but now I am traveling so it will take me a week or so.