Every convex set is of locally finite perimeter

Question

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

Answer 1

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.