The coarea formula states that for Lipschitz $u$ and open $\Omega\subset \mathbb{R}^n$, we have $$ \int_{\Omega} |\nabla u|dx = \int_{-\infty}^{+\infty} H^{n-1}(u^{-1}(t))dt $$ I am a little confused on why the dimensions of the LHS and RHS don't agree. Since $|\nabla u|$ has dimensions of $1/L$ where $L$ denotes unit length, the LHS has dimensions of $L^{n-1}$. On the other hand, since the $H^{n-1}$ is "basically" the surface measure in $\mathbb{R}^n$, it seems to have dimensions of $L^{n-1}$. After integrating over $t$, the RHS would then have dimensions of $L^n$.
Of course, all of this is not rigorous, but hopefully makes sense intuitively. I would just like to know why the formula diverges from intuition.
I realized that I was omitting something rather important and thus causing some misunderstanding of the coarea formula.
Let $u$ be of unit $A$ (where $A$ is a just a placeholder). Then $\nabla u$ has units of $A/L$ so that the LHS has unit of $AL^{n-1}$. On the RHS, since $u$ has units of $A$, the integration variable $t$ should also have units of $A$. Therefore, integrating the "surface area" which has units $L^{n-1}$ with respect to $dt$ causes the RHS to have units of $AL^{n-1}$.