I want to know if it is possible to have a general rule for approximating the perimeter of a set $E\subset \mathbb{R}^n$ with finite perimeter in terms of the volume (Lebesgue measure) of a sequence of related sets. I mean perimeter here in the sense used in the calculus of variations and geometric theory of measures. That is, the set $E\subset \mathbb{R}^n$ is a set with finite perimeter whenever its characteristic function $\chi_E$ belongs to $\mathrm{BV}$, where $\mathrm{BV}$ is the space of functions with bounded variation. In this case, there is a set $\partial^\ast E \subset \partial E$ called the reduced boundary of $E$; where $P(E) = H^{n-1} (\partial^\ast E)$. As a standard notation, $P$ stands for the perimeter, $H^{n-1}$ for the $(n-1)$-dimensional Hausdorff measure, and $| \cdot |$ for the $n$-dimensional Lebesgure measure.
A guess that comes to mind is to consider the sequence of neighborhoods $I_t (\partial^\ast E) $ of B reduced to the boundary; and then consider $|I_t(\partial^\ast E)|$ for very small $t>0$. More precisely, define $I_t$ as $$ I_t(\partial^\ast E)= \{x\in \mathbb{R}^n : \mathrm{dist}(x, \partial^\ast E) \leq t \}. $$ In this case, my specific question is whether the limit $$ \lim_{t \downarrow 0} \frac{|I_t(\partial^\ast E)|}{t} $$ approaches (at least to some extent) to the perimeter $P(E) = H^{n-1} (\partial^\ast E)$?
I tried to estimate the above limit using the co-area formula and so on; But I don't succeed. Any suggestions and comments including other formulas and rules are welcome.
I posted a similar question at stackexchange.
