# essential boundary

Let $E\subset\mathbf{R}^{n}$ be a measurable set. We define the essential boundary of $E$ as

 $\partial^{*}E:=\{x\in\mathbf{R}^{n}\colon 0<|E\cap B_{\rho}(x)|<|B_{\rho}(x)|,% \quad\forall\rho>0\}$

where $|\cdot|$ is the Lebesgue measure.

Compare the definition of $\partial^{*}E$ with the definition of the topological boundary $\partial E$ which can be written as

 $\partial E=\{x\in\mathbf{R}^{n}\colon\emptyset\subsetneq E\cap B_{\rho}(x)% \subsetneq B_{\rho}(x),\quad\forall\rho>0\}.$

Hence one clearly has $\partial^{*}E\subset\partial E$.

Notice that the essential boundary does not depend on the Lebesgue representative of the set $E$, in the sense that if $|E\triangle F|=0$ then $\partial^{*}E=\partial^{*}F$. For example if $E=\mathbf{Q}^{n}\subset\mathbf{R}^{n}$ is the set of points with rational coordinates, one has $\partial^{*}E=\emptyset$ while $\partial E=\mathbf{R}^{n}$.

Nevertheless one can easily prove that $\partial^{*}E$ is always a closed set (in the usual sense).

Title essential boundary EssentialBoundary 2013-03-22 15:01:54 2013-03-22 15:01:54 paolini (1187) paolini (1187) 9 paolini (1187) Definition msc 49-00