This is probably a simple question, maybe more suited for MSE. In the coarea formula, you have

$$\int_{{\mathbb{R}}^n} g (x) |\nabla f(x)|\, dx= \int_\mathbb{R} \left(\int_{\{f=t\}} g d \mathcal{H}^{n-1} \right)dt ,$$

with $f$ Lipshitz and $g$ Borel (positive maybe?). Anyway the question is: how do you define the integral $\int_{\{f=t\}} g d \mathcal{H}^{n-1} $? When you do the theory of Sobolev spaces it's often remarked that traces of $L^p$ functions make no sense, and here you're integrating generic measurable functions basically. I'm sure I (used to) know this but I am not seeing it right now.