I have a question about a integral on a surface.
It is well known that for any Integrable function $f$ defined on $\mathbb{R}^{n}$, it holds that \begin{equation} (1) \quad \frac{d}{dr} \int_{B(0,r)}f\,dm=\int_{\partial B(0,r)}f\,d \sigma \quad m\text{-a.e. }r. \end{equation} Here and hereafter $m$ denotes the $n$-dim Lebesgue measure, $\sigma$ the $(n-1)$ dim Hausdorff measure (surface measure) and $B(0,r)$ the open ball of radius $r$ centered at origin[See Lawrence C. Evans and Ronald E. Gariepy's book].
Question
Let $D \subset \mathbb{R}^{n}$ be a bounded domain with $C^{1}$ boundary. Set \begin{align} D_{\epsilon}=\left\{ x \in \bar{D} : d\left(x,\partial D \right) \leq \epsilon \right\} \end{align} Can we show the following equation? : \begin{align} \lim_{\epsilon \to 0} \frac{1}{\epsilon}\int_{D_{\epsilon}}f\,dm=\int_{\partial D}f\,d\sigma ,\quad (f \in C(\bar{D})) \end{align}
This is a generalization of $(1)$.
If you know how to prove this equation or helpful references, please let me know.
Thank you in advance.