By definition , when we are given a set $A \in \mathbb{R}^n$ , $ H_\delta^{n-1} (\partial A ) = \inf \left\{ \sum_{j=1}^{\infty} \alpha_{n-1}\frac{1}{2^{n-1}} [\operatorname{diam}(U_j)] ^{n-1} \mid \partial A \subseteq \cup U_j , \operatorname{diam}U_j \leq \delta \right\} ,$ $ H^{n-1} (\partial A ) = \lim_{\delta \to 0 } H_\delta^{n-1} (\partial A ).$
How can I prove that when I have $ \{A_i \},A $ which are open with smooth boundaries in $\mathbb{R}^n $ , such that $\operatorname{Leb}(A_i \Delta A ) \to 0 $ , then $ H^{n-1} ( \partial A) \leq \liminf_{ i \to \infty} H^{n-1} (\partial A_i )$ .
Thanks in advance !
p.s.- $\alpha_{n-1}$ is the volume of the $n-1$ dimensional unit ball