Htykuut

Before explaining my problem, I recall the definitions:

Let $E subset mathbb{R}^n$ be a Lebesgue measurable set. We say that $E$ is a set of locally finite perimeter if for every compact set $K subset mathbb{R}^n$ it holds that
begin{equation*} label{eq:deflocfiniteperimeter}
M_K := sup left{ int_{E} div, T(x) ,dx : T in C^1_c( mathbb{R}^n; mathbb{R}^n), spt ,T subset K, | T | leq 1 right} < infty.
end{equation*}
Moreover, if $$
supleft{M_K:K subset mathbb{R}^n, Ktext{compact}right}< infty,$$
then we say that $E$ is a set of finite perimeter.

Now suppose that $E$ is a set of finite perimeter. I have to prove that for $mathcal{L}^{n-1}$-a.e. $z in mathbb{R}^{n-1}$ the vertical slice $E_z subset mathbb{R}$ is a set of finite perimeter; where, by definition, $E_z := { t in mathbb{R} : (z,t ) in E}$.

I can only prove that for a.e. $z in mathbb{R}^{n-1}$ the vertical slice is a set of LOCALLY finite perimeter. Here I descrive my attempt:

Let $rho$ is a regulatizing kernel and consider the sequence $ u_{h} : = chi_E * rho_{1/h}$: I already know (from a previous result) that
$$ limsup_{h to infty} int_K |nabla u_h(x)| dx leq P(E;K)$$
for all compact sets $K subset mathbb{R}^{n}$. Now fix a compact $J subset mathbb{R}$. I have proved that if $T in C^1_c(mathbb{R})$ satisfies $|T| leq 1$ and $J supset spt T$, then for a.e. $z in mathbb{R}^{n-1}$ it holds
$$ left| int_{E_z} T'(t) , dt right| leq liminf_{h to infty} int_{J} |nabla u_h (z,t)| , dt .$$
Taking the sup among the functions $T in C^1_c(mathbb{R})$ with $|T| leq 1$ and $J supset spt T$ and integrating on a compact set $H subset mathbb{R}^{n-1}$, we get
$$ int_H sup left{ left| int_{E_z} T' right| : T in C^1_c(mathbb{R}), , |T| leq 1 , , J supset spt T right} leq liminf_{h to infty} int_{H times J} |nabla u_h| leq P(E; H times J ) leq P(E) < infty. $$
If the integral of a function is finite then the function is a.e. finite. Hence for a.e. $z$ we have
$$ M_{J} := sup left{ left| int_{E_z} T' right| : T in C^1_c(mathbb{R}), , |T| leq 1 , , J supset spt T right} < infty.$$
This proves that the set has locally finite perimeter but I can't find a uniform bound for $M_{J}$.

Any help would be really appreciated!

This is exactly the assertion of Vol'pert theorem

Theorem (Vol'pert) Let $E$ a set of finite perimeter in $mathbb R^n$ with $n>1$ then for $mathcal L^{n-1}$-a.e. points $x'inmathbb R^{n-1}$

• 
  $E_{x'}$ is a set of finite perimeter in $mathbb R$;


• 
  $partial^*(E_{x'})=(partial^* E)_{x'}$;


• 
  $nu^E_y(x', y)neq 0$ for every $(x', y)inpartial ^*(E_{x'})$ where $nu^E_y$ is the $n$-th component of the generalized inner normal of $E$ named $nu^E$.

A generalization of this theorem for $k$-dimensional slices where $k=1, 2, dotsc, n-1$ can be found at [1, Th. 2.93]

[1] Ambrosio, Luigi; Fusco, Nicola; Pallara, Diego, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. Oxford: Clarendon Press. xviii, 434 p. (2000). ZBL0957.49001.