The main thing i need to prove is the following assertion:
Let $E\subset R^N$ be a set of finite perimeter and $H=\{x\in R^N : x\cdot e < t \}$ for $t\in R$ and $e\in S^{N-1}$.
Then prove that $$ \text{Per}(E\cap H)\leq\text{Per}(E), $$ by using the following decomposition of the Gauss Green measure of $E\cap H$ $$ \mu_{E\cap H}= (\mu_{E})_{|_{H}} + e H^{N-1}_{|_{E\cap\partial H}}. $$
For the convenience of the reader i recall that E is of locally finite perimeter if there exists a vector-valued Radon measure $\mu_E$ such that $\int_{E} \text{div}Tdx=\int_{R^N} T\cdot d\mu_E$ for every smooth compactly supported vector field $T$, in this case $\text{Per}(E;A)=|\mu_E|(A)$.
The hint of the book is the following: Use the recalled decomposition with the vector field $T(x)=e$ for each $x$ (clearly a truncation argument is required).
My attempt is like the following, i consider $T_M(x)= \phi(|x|)e$ ($M>0$) where $\phi\in C^{\infty}(R)$ is such that $\phi=1$ on $(-\infty, M]$ and $\phi=0$ on $[M+1,\infty)$. Using this vector field in the decomposition and passing to the limit for $M\to\infty$ i can get easily
$$H^{N-1}(E\cap \partial H)\leq \text{Per}(E;H). $$
I can't fine out how to prove the derired inequality
