Sorry for asking such a basic question, but this is not my area of expertise.

In my work I'm using the coarea formula: for $\Omega \subseteq \mathbb{R}^n$ open and bounded, and $u : \Omega \to \mathbb{R}$ Lipschitz,

$\int_{\Omega} |\nabla u| = \int_{-\infty}^\infty \mathcal{H}_{n-1}(u^{-1}(t))dt$.

I can calculate the LHS and want to use this to reason about the RHS.

**Question:** Is it true that there is a "standard definition" of surface area such that for almost all $t$, the set $L_t = \{x \in \Omega : u(x) \geq t\}$ is "nice enough" that SurfaceArea$(L_t) = \mathcal{H}_{n-1}(u^{-1}(t))$? And is there a reasonable reference for this?

If it makes a difference, my $u$ is actually defined on the torus $\mathbb{R}^n/\mathbb{Z}^n$, which may make things cleaner.

Thanks!

`$\{u \geq t\}$`

would be`$\{u = t\}$`

. @Benoit Kloekcner: well... suppose it's merely Lipschitz -- am I out of luck? $\endgroup$