Theorem 6, pg 649 in appendix E of evans states the Lebesgue differentiation theorem:
THEOREM 6 (Lebesgue's Differentiation Theorem). Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be locally summable. Then for a.e. point $x_0 \in \mathbb{R}^n$, $$ \frac{1}{|B\left(x_0, r\right)|}\int_{B\left(x_0, r\right)} f d x \rightarrow f\left(x_0\right) \quad \text { as } r \rightarrow 0 . $$
I would like to know if its possible to extend it for a surface integral, that is, $$ \frac{1}{|\partial B\left(x_0, r\right)|}\int_{\partial B\left(x_0, r\right)} f(x) dS_x \rightarrow f\left(x_0\right) \quad \text { as } r \rightarrow 0 . $$ where $|\partial B\left(x_0, r\right)|$ denotes the surface area of the ball $B\left(x_0, r\right)$. This seems very intuitive, but I couldn't find any sources online. Please note that $f$ is not necessarily continuous.
Edit : Assume $f \in H^{1}(\overline{\Omega})$, where $\Omega = B\left(x_0, r\right)$.