Proof of relationship between Dirac Delta and Co-Area formula In the Wikipedia page for the Dirac Delta function this formula appears under "Properties in $n$ dimension".
$$
\int f(x) \delta(g(x)) dx = \int_{g^{-1}(0)} \frac{f(x)}{|\nabla g(x)|} d\sigma(x)
$$
It is said that this is a consequence of the Co-Area formula but no proof is given and the only reference ("Hörmander (1983), The analysis of linear partial differential operators I") doesn't seem to have this formula in it.
I have a few questions, in order of importance.


*

*What is a proof of this statement?

*What other references are there about this statement and its generalizations to a function $g:\mathbb{R}^n\to\mathbb{R}^m$ with $n > m > 1$?

*In the above the author uses $\delta (g(x)) dx$ as if $\delta$ was a function, where in fact it is a Schwartz distribution or a measure. What did they mean? Especially because now it is concatenated with another function.


Definition of Dirac Distribution
It's a linear functional that maps test functions $\varphi$ to
$$
\delta_x[\varphi] = \int \varphi(y) \delta_x^{\text{measure}}(dy) = \varphi(y)
$$
where $\delta_x^{\text{measure}}$ is the Dirac Measure which for any measurable set $A$ is defined as
$$
\delta_x^{\text{measure}}(A) = \begin{cases}
    1 & x\in A \\
    0 & x\notin A
\end{cases}
$$
Co-Area Formula for Lipschitz Functions
If $g:\mathbb{R}^n\to\mathbb{R}^m$ with $n > m$ then
$$
\int_{\mathbb{R}^n} f(x) dx = \int_{\mathbb{R}^m} \left[\int_{g^{-1}(y)} f(x) |J_g(x) J_g(x)^\top|^{-1/2} \mathcal{H}^{n-m}(dx) \right]dy 
$$
where $J_g(x)$ is the Jacobian matrix of $g$.
 A: They mean $$\int_{\Bbb{R}^N} f(x)\delta(g(x))dx=\lim_{n\to \infty}
\int_{\Bbb{R}^N} f(x)\frac{n}2 1_{|g(x)|< 1/n}dx$$ where $f\in C^\infty_c(\Bbb{R}^N)$ and $g\in C^\infty(\Bbb{R}^N)$ with $\|\nabla g\| \ne 0$.
What is $d\sigma(x)$? It is just the measure/distribution defined by
$$\int_{Z(g)} f(x)d\sigma(x) = \lim_{n\to \infty}
\frac{n}2\int_{dist(Z(g),x)<1/n} f(x)dx$$
where $Z(g)$ is the vanishing set of $g$ and $$dist(Z(g),x)=\inf_{a\in Z(g)} \|x-a\|$$
For $x$ close to $a\in Z(g)$ we have $g(x)\approx \nabla g(a) \cdot (x-a)$ from which $$dist(Z(g),x) \approx \frac{|g(x)|}{\|\nabla g(a)\|}$$
Whence for $f_a\in C^\infty_c(\Bbb{R}^N)$ supported on a small ball around $a$ we have $$\lim_{n\to \infty}
\int_{\Bbb{R}^N} f_a(x)\frac{n}2 1_{|g(x)|< 1/n}dx\approx 
\int_{Z(g)} \frac{f_a(x)}{\|\nabla g(a)\|}d\sigma(x)$$
Writing $f$ as a sum of smooth functions supported on very small balls  we get the formula
$$\int_{\Bbb{R}^N} f(x)\delta (g(x))dx=\int_{Z(g)} \frac{f(x)}{\|\nabla g(x)\|}d\sigma(x)$$
