# essential boundary

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

$$ |

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

$$\partial E=\{x\in {\mathbf{R}}^{n}:\mathrm{\varnothing}\u228aE\cap {B}_{\rho}(x)\u228a{B}_{\rho}(x),\forall \rho >0\}.$$ |

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 $|E\mathrm{\u25b3}F|=0$ 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 ${\partial}^{*}E=\mathrm{\varnothing}$ 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 |