Is there a standard analysis reference for the first formula given in these lecture notes:
$$\int_D h(x)\,dx = \int_{-\infty}^\infty dt\int_{D_f(t)}\frac{h(y)}{|\nabla f(y)}\,dS_y$$ where $dS_y$ is the surface measure on $D_f(t)=\{x\in D:f(x)=t\}$? Here $D\subset \mathbb R^n$ is a bounded domain, $f\in C^1(D)$, and $h\in C^0(D)$.
I have found only results that use "Hausdorff measure", but this is too complicated for me.