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
Canonical name	EssentialBoundary
Date of creation	2013-03-22 15:01:54
Last modified on	2013-03-22 15:01:54
Owner	paolini (1187)
Last modified by	paolini (1187)
Numerical id	9
Author	paolini (1187)
Entry type	Definition
Classification	msc 49-00