Averaging the mass of a Sobolev function $f\in W^{1,p}(\Omega)$ near $\partial\Omega$

In Evans & Gariepy's "Measure theory and fine properties of functions", Sec. 5.3., they construct the trace operator on a bounded Lipschitz domain $\Omega$ for BV-functions (and thus by inclusion for the subspaces $W^{1,p}$) in a similar fashion, using averages. In their case, they simply consider each part of the boundary locally as a graph $x_n = \gamma(x_1,...,x_{n-1})$ and then average vertically in the form $$Tf(x_1,...,x_{n-1}) = \lim_{r\to 0} \frac{1}{r}\int_0^r f(x_1,...,x_{n-1},\gamma(x_1,...,x_{n-1}))dr$$ but your construction should be close enough to allow for a similar proof.

They also prove the useful theorem 2 which says that for $\mathcal{H}^{n-1}$-almost all $x\in \partial \Omega$ we have $$\lim_{r\to 0} \frac{1}{|B_r(x) \cap \Omega|} \int_{B_r(x) \cap \Omega} |f-Tf(x)| dy = 0 $$ which also might be helpful.