We have a set of locally finite perimeter and a sequence of sets $\{E_h\}_h$ with $C^1$ boundary such that $$E_h\to E \text{ and } \mu_{E_h}\stackrel{*}{\rightharpoonup} \mu_E,$$ where $\mu_{E_h}$ and $\mu_E$ are the Gauss-Green measures of $E_h, E$ respectively.
We have already proved that for a.e. $r>0$
$\mu_{E_h\cap B_r}=\mu_{E_h}\lfloor B_r+ \mu_{B_r}\lfloor E_h \quad (*)$
($B_r$ is the ball of centre 0 and radius $r$) and we know also that $\mu_{B_r}\lfloor E_h\stackrel{*}{\rightharpoonup} \mu_{B_r}\lfloor E.$
We want now to prove that $\mu_{E_h\cap B_r}\stackrel{*}{\rightharpoonup} \mu_{E\cap B_r}$. We are reffering to pag. 173 of "sets of finite perimetr and geometrical variational problems", by F. Maggi, where there is written that from
- $(*)$;
- $\limsup_{h\to \infty} P(E_h\cap B_r)<\infty$;
- $E_h\cap B_r\to E\cap B_r$
the claim follows.
The problem is that we can't understand why this is sufficient to prove it, so any help will be really appreciated.
Proposition 12.15, p.126 in the book, is the answer to your question, and which the following is based on.
First, the fact that $\limsup_{h \to \infty} P(E_h\cap B_r) < \infty$ implies $E_h\cap B_r$ is a set of finite perimeter. Then we have \begin{equation} \int_{\mathbb{R}^n}\varphi\, d \mu_{E_h \cap B_r} = \int_{E_h \cap B_r}\nabla \varphi \,d x \quad \text{for all } \varphi \in C_c^\infty(\mathbb{R}^n). \end{equation} We also additionally know that $E_h\cap B_r \to E \cap B_r$, so \begin{equation} \lim_{h \to \infty}\int_{E_h \cap B_r}\nabla \varphi \,d x = \int_{E \cap B_r}\nabla \varphi \,d x = \int_{\mathbb{R}^n}\varphi\,d\mu_{E \cap B_r} \end{equation} i.e., \begin{equation} \lim_{h \to \infty}\int_{\mathbb{R}^n}\varphi\, d \mu_{E_h \cap B_r} = \int_{\mathbb{R}^n}\varphi\,d\mu_{E \cap B_r}\quad \text{for all } \varphi \in C_c^\infty(\mathbb{R}^n).\quad\quad (*) \end{equation} Now $C_c^\infty(\mathbb{R}^n)$ is dense in $C_c^{0}(\mathbb{R}^n)$ (with the supremum norm), which together with the fact that $E_h \cap B_r$ has finite perimeter, implies $(*)$ actually holds for all $\varphi \in C_c^0(\mathbb{R}^n)$. So $\mu_{E_h \cap B_r} \rightharpoonup^* \mu_{E\cap B_r}$.