I thought you were going to ask that! My first attempt had a $\rho$ taylor to the norm, and the argument $f = 1$ in $A$ and $0$ outside $A^{\epsilon}$ could be done using balls in the p-norm. But I thought that the bounds on derivatives of $f$ would involve $n.$ May be you or someone can check, and see if there is a way out. I'll add the construction of the $\rho$ as answer.

[Wikipedia][1] mentions Federer's book. The results is that there is a Borel set $S_0$ contained in your set $S$ such that $\nu(S_0^c)=0$, since $S^c \subset S_0^c$, then $S = S_0 \cup \text{(a set contained in a $\nu$ null set})$ [1]: en.wikipedia.org/wiki/Lebesgue_differentiation_theorem

$\lambda$ can be extended to the sigma algebra generated by Borel sets and $\nu$ null sets. The simplest way is to extend $\lambda$ as $\lambda^{ \star}(A) = \int_A L(y) \space d\nu(y) $.