I need to prove that every convex subset of $\mathbb{R}^n$ is of locally finite perimeter. 


$E$ is of locally finite perimeter if there exists a vector-valued Radon measure $\mu\_E$ s.t. the Gauss Green theorem holds: that is for each compactly supported vector field $T$
$$ \int\_{E}div(T)=\int\_{\mathbb{R}^n}T\cdot d\mu\_E. $$


Moreover the perimeter of $E$ is defined as the total variation of $\mu\_E$, that is $P(E;A):=|\mu\_E|(A)$.


Let me state the following lemma which the book suggests to use: 
Let $H\_t=\{x : e\cdot x<t \} $ for $t\in \mathbb{R}$ and $e\in S^{n-1}$ (an half space) and $E$ a set of locally finite perimeter with $|E|<\infty $. Then 
$$ \mu\_{E\cap H\_t}= (\mu\_E)\_{|\_{H\_t}}+ eH^{n-1}\_{|\_{E\cap H\_t}}. $$
From this follow that $H^{n-1}(E\cap \partial H\_t)\leq P(E; H\_t)$, $P(E\cap H\_t)\leq P(E)=P(E;\mathbb{R}^n)$.


Let now $C$ be a convex set, this happens if and only if $\bar{C}=\bigcap\_n H\_n$ where $H\_n$ are closed half spaces. 
The suggestion from the book is the following: first prove that if $E$ is of finite perimeter and $C$ is convex, then $P(E\cap C)\leq P(E)$ (which is an easy consequence of the second inequality in the last result and of the fact that $C$ is a countable intersection of half spaces). Then refine this argument to prove that every convex set if of locally finite perimeter. 


I didn't get the suggestion of "refine" the argument, and so i am asking for help. 
Thanks to everyone who will use time to respond me 



Every convex set is of locally finite perimeter
First assume that $E$ is compact. Then, your inequality says that you can approximate it from above by a sequence $E\_n$ of convex polytopes with decreasing perimeters. Then, the sequence $\mu\_{E\_n}$ is weak\*-precompact by Banach$-$Alaoglu theorem, so we can assume by passing to a subsequence that $\mu\_{E\_n}\to \mu$. Hence 
$$
\int\_E \mathrm{div}\,T=\lim\_{n\to\infty} 
\int\_{E\_n} \mathrm{div}\,T=\lim\_{n\to\infty} \int Td\mu\_{E\_n}=\int Td\mu,
$$ and we are done.


If $E$ is not compact, intersect it with balls of large radii and and pass to a limit.


