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 Lebesguerepresentative of the set $E$ , in the sense that if $|E\\triangle F|=0$ then $\\partial^{*}E=\\partial^{*}F$ . For example if $E=\\mathbf{Q}^{n}\\subset\\mathbf{R}^{n}$ isthe 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).