# Hausdorff measure of intersection of a ball and a set in $\mathbb {R} ^ n$

Let $$A$$ a subset of $$\mathbb R ^n$$, $$B=B(x,r) \subset \mathbb {R} ^n$$ an open ball, and denote the $$(n-1)$$-dimensional Hausdorff measure in $$\mathbb R ^n$$ by $$\mathcal H^{n-1}$$. Also assume that $$\mathcal H^{n-1} (\partial A) < + \infty$$ (One can assume that $$A$$ is a set of finite perimeter in necessary).

In this case, is it the following identity holds?

$$\mathcal H^{n-1}(\partial (A \cap B))= \mathcal H^{n-1}((\partial A) \cap B) + \mathcal H^{n-1}( A\cap ( \partial B))$$

Let me state and prove the following:

Proposition. Let $$E \subset \mathbb R^n$$ be a set of finite perimeter. For $$\mathcal L^1$$-a.e. $$\rho>0$$ the following equality holds: $$P(E \cap B_{\rho}) = P(E, B_{\rho}) + \mathcal H^{n-1}(E \cap \partial B_{\rho}).$$

Proof. Let $$u \in BV(\mathbb R^n)$$ and set $$v_\rho:=u \chi_{B_{\rho}}$$, being $$\chi_A$$ the characteristic function of a set $$A$$. Then we have $$\tag{0} Dv_{\rho} = (Du)\llcorner_{B_\rho} - \gamma_{\rho}(u) \cdot \nu \mathcal H^{n-1}\llcorner_{\partial B_{\rho}}$$ where $$\gamma_{\rho}(u)$$ is the trace of $$u$$ on $$B_\rho$$. By a well-known characterization of the trace operator, we know that if $$x \in \partial B_\rho$$ $$\tag{1} \lim_{r \to 0^+} \frac{1}{r^n} \int_{B_\rho \cap B_r(x)} \vert u(y)-\gamma_{\rho}(u)(x) \vert \, dy \to 0, \qquad \mathcal H^{n-1}\text{-a.e. } x \in \partial B_\rho.$$ On the other hand, we know that $$\mathcal L^n$$-a.e. $$x$$ is a Lebesgue point for $$u$$, i.e. $$\tag{2} \lim_{r \to 0^+} \frac{1}{r^n} \int_{B_r(x)} \vert u(y)- u(x) \vert \, dy \to 0, \qquad \mathcal L^{n}\text{-a.e. } x \in \mathbb R^n.$$ Furthermore, by Coarea Formula, we know that if $$f$$ is a Lipschitz map $$\vert \nabla f \vert \mathcal L^n = \mathcal H^{n-1}\llcorner_{\{f=t\}} \otimes \mathcal L^1.$$ Now take $$f=\vert \cdot \vert$$ and hence $$\begin{equation*} \mathcal L^n = \mathcal H^{n-1}\llcorner_{\partial B_{\rho}} \otimes \mathcal L^1, \end{equation*}$$ (which is the well-known formula for the polar change of coordinates). From this it follows that (2) is equivalent to $$\text{for } \mathcal L^1\text{-a.e. } \rho>0, \, \lim_{r \to 0^+} \frac{1}{r^n} \int_{B_r(x)} \vert u(y)- u(x) \vert \, dy \to 0, \quad \mathcal H^{n-1}\text{-a.e. } x \in \partial B_\rho.$$ Hence, being the integrand non negative, we also deduce
$$\tag{3} \text{for } \mathcal L^1\text{- a.e. } \rho>0, \lim_{r \to 0^+} \frac{1}{r^n} \int_{B_r(x) \cap B_\rho} \vert u(y)- u(x) \vert \, dy \to 0, \quad \mathcal H^{n-1}\text{-a.e. } x \in \partial B_\rho.$$ By comparison between (1) and (3) we get $$\begin{equation*} \text{for } \mathcal L^1\text{ - a.e. } \rho>0, \quad \gamma_\rho(u) = u \vert_{\partial B \rho}, \quad \mathcal H^{n-1}\text{-a.e. } x \in \partial B_\rho. \end{equation*}$$ Now it is easy to conclude: from (0) we have $$Dv_{\rho} = (Du)\llcorner_{B_\rho} - u \cdot \nu \mathcal H^{n-1}\llcorner_{\partial B_{\rho}}, \qquad \mathcal L^1 \text{-a.e. } \rho >0$$ which is $$D(\chi_{E\cap B_{\rho}})= (D\chi_E)\llcorner_{B_\rho} - \chi_E \cdot \nu \mathcal H^{n-1}\llcorner_{\partial B_{\rho}}, \qquad \mathcal L^1 \text{-a.e. } \rho >0$$ and since the two measures are mutually singular we deduce, taking total variations, $$\vert D(\chi_{E\cap B_{\rho}})\vert = \vert D\chi_E \vert \llcorner_{B_\rho} + \mathcal H^{n-1}\llcorner_{(\partial B_{\rho} \cap E)}, \qquad \mathcal L^1 \text{-a.e. } \rho >0.$$

• Thanks! It really make sense. – XIE Jan 24 '19 at 15:18
• @HamedPourmohammad You are welcome, I am glad to help when I can. – Romeo Jan 24 '19 at 15:56
• just two (maybe stupid) questions: 1- $E$ is a set of finite perimeter, then by De Giorgi's structure theorem, $P(E)=\mathcal H ^{n-1}(\partial ^\ast E)$. Am I right? 2- I understand that your statement is true for a.e. $\rho$. But fix a ball $B=B_{\rho}$ and let $E= \mathbb {R} ^ n \setminus B$. Then for this $E$ and $B$ the both statements are false. Is it possible to improve it? – XIE Jan 28 '19 at 11:48
• Question 1. Yes. Question 2. I have not understood which are the statements you are referring to. – Romeo Jan 28 '19 at 13:01