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!



Technical question on perimeter of level sets
The answer to your first question is yes. You can take as surface area of a set the (total) variation of its characteristic function. This is the "standard definition" of [perimeter](http://en.wikipedia.org/wiki/Caccioppoli_set#Caccioppoli_definition) for general measurable sets in $\mathbb{R}^n$. See, for example, [this book](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=AUCN&pg7=ALLF&pg8=ET&review_format=html&s4=ambrosio&s5=fusco&s6=pallara&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=1&mx-pid=1857292), in particular Definition 3.35 and Theorem 3.40.


