Prove that a sequence of measures weak-star converges to another measure 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.
 A: 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 $\mu_{E_h \cap B_r} \rightharpoonup^* \mu_{E\cap B_r}$.
