We have the following definition of perimeter, freely available from your favourite text on geometric measure theory:
$P(E)=\sup\bigg\{\int_E \nabla \cdot \varphi(x)\,dx:\varphi\in C^1_c(\mathbb{R}^n;\mathbb{R}^n):|\varphi|\leq 1\bigg\}$
My question is, why not define perimeter using $\varphi\in C^1_c(\mathbb{R}^n;\mathbb{R}^n)-a.e.$? We are constantly defining things up to null sets in geometric measure theory and this surely makes it easier when wanting to demonstrate the definitions validity for $C^1$-rectifiable sets. If $\nu_E(x)$ is the outer unit normal at $x$, then for any $C^1$-rectifiable set $E$ we can just let $\varphi(x)=\nu_E(x)$ where it exists and $0$ else. Then a straightforward application of the Gauss-Green/Divergence theorem gets us where we need to go.
Is it purely convention that perimeter is not defined this way or is there some subtlety that I am missing? Perhaps there is a subsequent definition/theorem that is more cumbersome if we include the $-a.e.$?